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Theorem opabiotafun 6448
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 6103 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦𝜑} = {𝑦})
2 mo2icl 3544 . . . . 5 (∀𝑧({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑}) → ∃*𝑧{𝑦𝜑} = {𝑧})
3 unieq 4602 . . . . . 6 ({𝑦𝜑} = {𝑧} → {𝑦𝜑} = {𝑧})
4 vex 3353 . . . . . . 7 𝑧 ∈ V
54unisn 4610 . . . . . 6 {𝑧} = 𝑧
63, 5syl6req 2816 . . . . 5 ({𝑦𝜑} = {𝑧} → 𝑧 = {𝑦𝜑})
72, 6mpg 1892 . . . 4 ∃*𝑧{𝑦𝜑} = {𝑧}
8 nfv 2009 . . . . 5 𝑧{𝑦𝜑} = {𝑦}
9 nfab1 2909 . . . . . 6 𝑦{𝑦𝜑}
109nfeq1 2921 . . . . 5 𝑦{𝑦𝜑} = {𝑧}
11 sneq 4344 . . . . . 6 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211eqeq2d 2775 . . . . 5 (𝑦 = 𝑧 → ({𝑦𝜑} = {𝑦} ↔ {𝑦𝜑} = {𝑧}))
138, 10, 12cbvmo 2627 . . . 4 (∃*𝑦{𝑦𝜑} = {𝑦} ↔ ∃*𝑧{𝑦𝜑} = {𝑧})
147, 13mpbir 222 . . 3 ∃*𝑦{𝑦𝜑} = {𝑦}
151, 14mpgbir 1894 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
16 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
1716funeqi 6089 . 2 (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
1815, 17mpbir 222 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  ∃*wmo 2563  {cab 2751  {csn 4334   cuni 4594  {copab 4871  Fun wfun 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-fun 6070
This theorem is referenced by:  opabiota  6450
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