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Theorem opabiotafun 6217
Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.)
Hypothesis
Ref Expression
opabiota.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
Assertion
Ref Expression
opabiotafun Fun 𝐹
Distinct variable group:   𝑥,𝑦,𝐹
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabiotafun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 funopab 5883 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦𝜑} = {𝑦})
2 mo2icl 3372 . . . . 5 (∀𝑧({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑}) → ∃*𝑧{𝑦𝜑} = {𝑧})
3 unieq 4415 . . . . . 6 ({𝑦𝜑} = {𝑧} → {𝑦𝜑} = {𝑧})
4 vex 3194 . . . . . . 7 𝑧 ∈ V
54unisn 4422 . . . . . 6 {𝑧} = 𝑧
63, 5syl6req 2677 . . . . 5 ({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑})
72, 6mpg 1721 . . . 4 ∃*𝑧{𝑦𝜑} = {𝑧}
8 nfv 1845 . . . . 5 𝑧{𝑦𝜑} = {𝑦}
9 nfab1 2769 . . . . . 6 𝑦{𝑦𝜑}
109nfeq1 2780 . . . . 5 𝑦{𝑦𝜑} = {𝑧}
11 sneq 4163 . . . . . 6 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211eqeq2d 2636 . . . . 5 (𝑦 = 𝑧 → ({𝑦𝜑} = {𝑦} ↔ {𝑦𝜑} = {𝑧}))
138, 10, 12cbvmo 2510 . . . 4 (∃*𝑦{𝑦𝜑} = {𝑦} ↔ ∃*𝑧{𝑦𝜑} = {𝑧})
147, 13mpbir 221 . . 3 ∃*𝑦{𝑦𝜑} = {𝑦}
151, 14mpgbir 1723 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
16 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
1716funeqi 5870 . 2 (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
1815, 17mpbir 221 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  ∃*wmo 2475  {cab 2612  {csn 4153   cuni 4407  {copab 4677  Fun wfun 5844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-fun 5852
This theorem is referenced by:  opabiota  6219
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