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Theorem opabiotafun 6169
 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 5837 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦𝜑} = {𝑦})
2 mo2icl 3352 . . . . 5 (∀𝑧({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑}) → ∃*𝑧{𝑦𝜑} = {𝑧})
3 unieq 4380 . . . . . 6 ({𝑦𝜑} = {𝑧} → {𝑦𝜑} = {𝑧})
4 vex 3176 . . . . . . 7 𝑧 ∈ V
54unisn 4387 . . . . . 6 {𝑧} = 𝑧
63, 5syl6req 2661 . . . . 5 ({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑})
72, 6mpg 1715 . . . 4 ∃*𝑧{𝑦𝜑} = {𝑧}
8 nfv 1830 . . . . 5 𝑧{𝑦𝜑} = {𝑦}
9 nfab1 2753 . . . . . 6 𝑦{𝑦𝜑}
109nfeq1 2764 . . . . 5 𝑦{𝑦𝜑} = {𝑧}
11 sneq 4135 . . . . . 6 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211eqeq2d 2620 . . . . 5 (𝑦 = 𝑧 → ({𝑦𝜑} = {𝑦} ↔ {𝑦𝜑} = {𝑧}))
138, 10, 12cbvmo 2494 . . . 4 (∃*𝑦{𝑦𝜑} = {𝑦} ↔ ∃*𝑧{𝑦𝜑} = {𝑧})
147, 13mpbir 220 . . 3 ∃*𝑦{𝑦𝜑} = {𝑦}
151, 14mpgbir 1717 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
16 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
1716funeqi 5824 . 2 (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
1815, 17mpbir 220 1 Fun 𝐹
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∃*wmo 2459  {cab 2596  {csn 4125  ∪ cuni 4372  {copab 4642  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by:  opabiota  6171
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