The source of the proof that Vishnu is the Supreme Lord of the Universe and is responsible for janma, sthithi, laya and others is in the Shastra. And in Shastra alone is the pramana.
The four vEdas, moola ramayaNa, mahAbhArata, pancharAtrAgama and the satvika purANAs are the pramANA shAstras. Other works and portions of other works that are in line with these pramANAs are also acceptable shAstras.
Vedavit: One who is obtained through the Vedas. vEdEna vidyatE labhyatE iti vEdavit
Vaidya: One who is propagated by all the (satvika) Vidyas. pratipAdyatayA sakala vidyA sambandhitvAdvaidyaH
Brahmana: One who is known through the Vedas. ‘brahmaNA vEdEna gamyata iti brAhmaNashabdO vishNAvEva yujyatE…. – nyAya vivaraNa of sri madhwAchArya’
Chaturvedavit: One who is obtained through the four Vedas. chaturvEdaiH labhyata ti chaturvEdavit.
Pramanam: One for revealing whom the Vedas and other pramanas (proofs) exist. prakrushTaM mAnaM vEdAdipramANaM yasminniti pramANaM.
Hence the purport of shAstrayOnitvAdhikaraNaM is covered by the Vishnusahasranama.