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Theorem opabiotafun 6421
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 6084 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦𝜑} = {𝑦})
2 mo2icl 3526 . . . . 5 (∀𝑧({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑}) → ∃*𝑧{𝑦𝜑} = {𝑧})
3 unieq 4596 . . . . . 6 ({𝑦𝜑} = {𝑧} → {𝑦𝜑} = {𝑧})
4 vex 3343 . . . . . . 7 𝑧 ∈ V
54unisn 4603 . . . . . 6 {𝑧} = 𝑧
63, 5syl6req 2811 . . . . 5 ({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑})
72, 6mpg 1873 . . . 4 ∃*𝑧{𝑦𝜑} = {𝑧}
8 nfv 1992 . . . . 5 𝑧{𝑦𝜑} = {𝑦}
9 nfab1 2904 . . . . . 6 𝑦{𝑦𝜑}
109nfeq1 2916 . . . . 5 𝑦{𝑦𝜑} = {𝑧}
11 sneq 4331 . . . . . 6 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211eqeq2d 2770 . . . . 5 (𝑦 = 𝑧 → ({𝑦𝜑} = {𝑦} ↔ {𝑦𝜑} = {𝑧}))
138, 10, 12cbvmo 2644 . . . 4 (∃*𝑦{𝑦𝜑} = {𝑦} ↔ ∃*𝑧{𝑦𝜑} = {𝑧})
147, 13mpbir 221 . . 3 ∃*𝑦{𝑦𝜑} = {𝑦}
151, 14mpgbir 1875 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
16 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
1716funeqi 6070 . 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 1632  ∃*wmo 2608  {cab 2746  {csn 4321   cuni 4588  {copab 4864  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-fun 6051
This theorem is referenced by:  opabiota  6423
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