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Theorem opabiota 6737
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 6661 . . 3 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
2 opabiota.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32iotabidv 6327 . . 3 (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
41, 3eqeq12d 2840 . 2 (𝑥 = 𝐵 → ((𝐹𝑥) = (℩𝑦𝜑) ↔ (𝐹𝐵) = (℩𝑦𝜓)))
5 vex 3483 . . . 4 𝑥 ∈ V
65eldm 5756 . . 3 (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7 nfiota1 6304 . . . . 5 𝑦(℩𝑦𝜑)
87nfeq2 2999 . . . 4 𝑦(𝐹𝑥) = (℩𝑦𝜑)
9 opabiota.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
109opabiotafun 6735 . . . . . 6 Fun 𝐹
11 funbrfv 6707 . . . . . 6 (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
1210, 11ax-mp 5 . . . . 5 (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦)
13 df-br 5053 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
149eleq2i 2907 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}})
15 opabidw 5399 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} ↔ {𝑦𝜑} = {𝑦})
1613, 14, 153bitri 300 . . . . . . 7 (𝑥𝐹𝑦 ↔ {𝑦𝜑} = {𝑦})
17 vsnid 4587 . . . . . . . . 9 𝑦 ∈ {𝑦}
18 id 22 . . . . . . . . 9 ({𝑦𝜑} = {𝑦} → {𝑦𝜑} = {𝑦})
1917, 18eleqtrrid 2923 . . . . . . . 8 ({𝑦𝜑} = {𝑦} → 𝑦 ∈ {𝑦𝜑})
20 abid 2806 . . . . . . . 8 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
2119, 20sylib 221 . . . . . . 7 ({𝑦𝜑} = {𝑦} → 𝜑)
2216, 21sylbi 220 . . . . . 6 (𝑥𝐹𝑦𝜑)
23 vex 3483 . . . . . . . . 9 𝑦 ∈ V
245, 23breldm 5764 . . . . . . . 8 (𝑥𝐹𝑦𝑥 ∈ dom 𝐹)
259opabiotadm 6736 . . . . . . . . 9 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
2625abeq2i 2951 . . . . . . . 8 (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
2724, 26sylib 221 . . . . . . 7 (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28 iota1 6320 . . . . . . 7 (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
2927, 28syl 17 . . . . . 6 (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
3022, 29mpbid 235 . . . . 5 (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
3112, 30eqtr4d 2862 . . . 4 (𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
328, 31exlimi 2219 . . 3 (∃𝑦 𝑥𝐹𝑦 → (𝐹𝑥) = (℩𝑦𝜑))
336, 32sylbi 220 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (℩𝑦𝜑))
344, 33vtoclga 3560 1 (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wex 1781  wcel 2115  ∃!weu 2654  {cab 2802  {csn 4550  cop 4556   class class class wbr 5052  {copab 5114  dom cdm 5542  cio 6300  Fun wfun 6337  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351
This theorem is referenced by: (None)
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