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Theorem fvopab5 6270
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 3201 . 2 (𝐴𝑉𝐴 ∈ V)
2 df-fv 5860 . . . 4 (𝐹𝐴) = (℩𝑧𝐴𝐹𝑧)
3 breq2 4622 . . . . 5 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
4 nfcv 2761 . . . . . 6 𝑦𝐴
5 fvopab5.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 nfopab2 4687 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
75, 6nfcxfr 2759 . . . . . 6 𝑦𝐹
8 nfcv 2761 . . . . . 6 𝑦𝑧
94, 7, 8nfbr 4664 . . . . 5 𝑦 𝐴𝐹𝑧
10 nfv 1840 . . . . 5 𝑧 𝐴𝐹𝑦
113, 9, 10cbviota 5820 . . . 4 (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
122, 11eqtri 2643 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
13 nfcv 2761 . . . . . . 7 𝑥𝐴
14 nfopab1 4686 . . . . . . . 8 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
155, 14nfcxfr 2759 . . . . . . 7 𝑥𝐹
16 nfcv 2761 . . . . . . 7 𝑥𝑦
1713, 15, 16nfbr 4664 . . . . . 6 𝑥 𝐴𝐹𝑦
18 nfv 1840 . . . . . 6 𝑥𝜓
1917, 18nfbi 1830 . . . . 5 𝑥(𝐴𝐹𝑦𝜓)
20 breq1 4621 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
21 fvopab5.2 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2220, 21bibi12d 335 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝜑) ↔ (𝐴𝐹𝑦𝜓)))
23 df-br 4619 . . . . . 6 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
245eleq2i 2690 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25 opabid 4947 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2623, 24, 253bitri 286 . . . . 5 (𝑥𝐹𝑦𝜑)
2719, 22, 26vtoclg1f 3254 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑦𝜓))
2827iotabidv 5836 . . 3 (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
2912, 28syl5eq 2667 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑦𝜓))
301, 29syl 17 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  Vcvv 3189  cop 4159   class class class wbr 4618  {copab 4677  cio 5813  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-iota 5815  df-fv 5860
This theorem is referenced by:  ajval  27587  adjval  28619
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