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Theorem fvopab5 6559
Description: The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvopab5.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
fvopab5.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
fvopab5 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem fvopab5
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3430 . 2 (𝐴𝑉𝐴 ∈ V)
2 df-fv 6132 . . . 4 (𝐹𝐴) = (℩𝑧𝐴𝐹𝑧)
3 breq2 4878 . . . . 5 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
4 nfcv 2970 . . . . . 6 𝑦𝐴
5 fvopab5.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 nfopab2 4944 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
75, 6nfcxfr 2968 . . . . . 6 𝑦𝐹
8 nfcv 2970 . . . . . 6 𝑦𝑧
94, 7, 8nfbr 4921 . . . . 5 𝑦 𝐴𝐹𝑧
10 nfv 2015 . . . . 5 𝑧 𝐴𝐹𝑦
113, 9, 10cbviota 6092 . . . 4 (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
122, 11eqtri 2850 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
13 nfcv 2970 . . . . . . 7 𝑥𝐴
14 nfopab1 4943 . . . . . . . 8 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
155, 14nfcxfr 2968 . . . . . . 7 𝑥𝐹
16 nfcv 2970 . . . . . . 7 𝑥𝑦
1713, 15, 16nfbr 4921 . . . . . 6 𝑥 𝐴𝐹𝑦
18 nfv 2015 . . . . . 6 𝑥𝜓
1917, 18nfbi 2008 . . . . 5 𝑥(𝐴𝐹𝑦𝜓)
20 breq1 4877 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
21 fvopab5.2 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2220, 21bibi12d 337 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝜑) ↔ (𝐴𝐹𝑦𝜓)))
23 df-br 4875 . . . . . 6 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
245eleq2i 2899 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25 opabid 5209 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2623, 24, 253bitri 289 . . . . 5 (𝑥𝐹𝑦𝜑)
2719, 22, 26vtoclg1f 3482 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑦𝜓))
2827iotabidv 6108 . . 3 (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
2912, 28syl5eq 2874 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑦𝜓))
301, 29syl 17 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1658  wcel 2166  Vcvv 3415  cop 4404   class class class wbr 4874  {copab 4936  cio 6085  cfv 6124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-iota 6087  df-fv 6132
This theorem is referenced by:  ajval  28273  adjval  29305
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