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Theorem opabiota 6985
Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
Hypotheses
Ref Expression
opabiota.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
opabiota.2 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
opabiota (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem opabiota
StepHypRef Expression
1 fveq2 6901 . . 3 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
2 opabiota.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32iotabidv 6538 . . 3 (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
41, 3eqeq12d 2742 . 2 (𝑥 = 𝐵 → ((𝐹𝑥) = (℩𝑦𝜑) ↔ (𝐹𝐵) = (℩𝑦𝜓)))
5 vex 3466 . . . 4 𝑥 ∈ V
65eldm 5907 . . 3 (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7 nfiota1 6508 . . . . 5 𝑦(℩𝑦𝜑)
87nfeq2 2910 . . . 4 𝑦(𝐹𝑥) = (℩𝑦𝜑)
9 opabiota.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
109opabiotafun 6983 . . . . . 6 Fun 𝐹
11 funbrfv 6952 . . . . . 6 (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
1210, 11ax-mp 5 . . . . 5 (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦)
13 df-br 5154 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
149eleq2i 2818 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
15 opabidw 5530 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ {𝑦𝜑} = {𝑦})
1613, 14, 153bitri 296 . . . . . . 7 (𝑥𝐹𝑦 ↔ {𝑦𝜑} = {𝑦})
17 vsnid 4670 . . . . . . . . 9 𝑦 ∈ {𝑦}
18 id 22 . . . . . . . . 9 ({𝑦𝜑} = {𝑦} → {𝑦𝜑} = {𝑦})
1917, 18eleqtrrid 2833 . . . . . . . 8 ({𝑦𝜑} = {𝑦} → 𝑦 ∈ {𝑦𝜑})
20 abid 2707 . . . . . . . 8 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
2119, 20sylib 217 . . . . . . 7 ({𝑦𝜑} = {𝑦} → 𝜑)
2216, 21sylbi 216 . . . . . 6 (𝑥𝐹𝑦𝜑)
23 vex 3466 . . . . . . . . 9 𝑦 ∈ V
245, 23breldm 5915 . . . . . . . 8 (𝑥𝐹𝑦𝑥 ∈ dom 𝐹)
259opabiotadm 6984 . . . . . . . . 9 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
2625eqabri 2870 . . . . . . . 8 (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
2724, 26sylib 217 . . . . . . 7 (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28 iota1 6531 . . . . . . 7 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
2927, 28syl 17 . . . . . 6 (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
3022, 29mpbid 231 . . . . 5 (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
3112, 30eqtr4d 2769 . . . 4 (𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
328, 31exlimi 2206 . . 3 (∃𝑦 𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
336, 32sylbi 216 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (℩𝑦𝜑))
344, 33vtoclga 3558 1 (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2557  {cab 2703  {csn 4633  cop 4639   class class class wbr 5153  {copab 5215  dom cdm 5682  cio 6504  Fun wfun 6548  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562
This theorem is referenced by: (None)
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