In his Kṛṣṇa-sandarbha (Anuccheda 87), Śrīla Jīva Gosvāmī cites the following verse of the Gopāla-tāpanī Upaniṣad (2.40) to prove that the Pradyumna who is the son of Kṛṣṇa and Rukmiṇī is the same Pradyumna who is a member of Lord Kṛṣṇa’s eternal fourfold plenary expansion, the catur-vyūha:

yatrāsau saṁsthitaḥ kṛṣṇas
 tribhiḥ śaktyā samāhitaḥ
rāmāniruddha-pradyumnai
 rukmiṇyā sahito vibhuḥ

“There [in Dvārakā] the almighty Lord Kṛṣṇa, endowed with His full potency, resided in the company of His three plenary expansions — Balarāma, Aniruddha and Pradyumna.” The Kṛṣṇa-sandarbha goes on to explain, with reference to the present verse of the Śrīmad-Bhāgavatam, that “the Cupid whom Rudra burned up with his anger is a demigod subordinate to Indra. This demigod Cupid is a partial manifestation of the prototype Cupid, Pradyumna, who is a plenary expansion of Vāsudeva. The demigod Cupid, being unable to attain a new body on his own, entered within the body of Pradyumna. Otherwise Cupid would have had to remain in a perpetual state of disembodiment, a result of Rudra’s having incinerated him with his anger.”

In his English rendering of the Śrīmad-Bhāgavatam (1.14.30 purport), Śrīla Prabhupāda confirms the absolute status of Pradyumna, Lord Kṛṣṇa’s first son: “Pradyumna and Aniruddha are also expansions of the Personality of Godhead, and thus They are also viṣṇu-tattva. At Dvārakā Lord Vāsudeva is engaged in His transcendental pastimes along with His plenary expansions, namely Saṅkarṣaṇa, Pradyumna and Aniruddha, and therefore each and every one of Them can be addressed as the Personality of Godhead, …”

According to Śrīla Śrīdhara Svāmī, Pradyumna took birth from the womb of Rukmiṇī before Śrī Kṛṣṇa’s marriage to Jāmbavatī and the Lord’s other marriages took place. Subsequently, Pradyumna returned from Śambara’s palace. But before Śukadeva Gosvāmī tells of Kṛṣṇa’s pastimes with His other wives, he will narrate the entire story of Pradyumna for the sake of continuity.

Śrīla Śrīdhara Svāmī further notes that Kāmadeva, or Cupid, now appearing within Pradyumna, is a portion of Vāsudeva because he is manifest from the element citta, consciousness, which is presided over by Vāsudeva, and also because he (Cupid) is the cause of material generation. As the Lord states in the Bhagavad-gītā (10.28), prajanaś cāsmi kandarpaḥ: “Of progenitors I am Kandarpa [Cupid].”