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Theorem opabiotafun 6972
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 6583 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦𝜑} = {𝑦})
2 mo2icl 3710 . . . . 5 (∀𝑧({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑}) → ∃*𝑧{𝑦𝜑} = {𝑧})
3 unieq 4919 . . . . . 6 ({𝑦𝜑} = {𝑧} → {𝑦𝜑} = {𝑧})
4 unisnv 4931 . . . . . 6 {𝑧} = 𝑧
53, 4eqtr2di 2788 . . . . 5 ({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑})
62, 5mpg 1798 . . . 4 ∃*𝑧{𝑦𝜑} = {𝑧}
7 nfv 1916 . . . . 5 𝑧{𝑦𝜑} = {𝑦}
8 nfab1 2904 . . . . . 6 𝑦{𝑦𝜑}
98nfeq1 2917 . . . . 5 𝑦{𝑦𝜑} = {𝑧}
10 sneq 4638 . . . . . 6 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1110eqeq2d 2742 . . . . 5 (𝑦 = 𝑧 → ({𝑦𝜑} = {𝑦} ↔ {𝑦𝜑} = {𝑧}))
127, 9, 11cbvmow 2596 . . . 4 (∃*𝑦{𝑦𝜑} = {𝑦} ↔ ∃*𝑧{𝑦𝜑} = {𝑧})
136, 12mpbir 230 . . 3 ∃*𝑦{𝑦𝜑} = {𝑦}
141, 13mpgbir 1800 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
15 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
1615funeqi 6569 . 2 (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
1714, 16mpbir 230 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ∃*wmo 2531  {cab 2708  {csn 4628   cuni 4908  {copab 5210  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545
This theorem is referenced by:  opabiota  6974
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