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Theorem fvopab5 6985
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 3466 . 2 (𝐴𝑉𝐴 ∈ V)
2 df-fv 6509 . . . 4 (𝐹𝐴) = (℩𝑧𝐴𝐹𝑧)
3 breq2 5114 . . . . 5 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
4 nfcv 2908 . . . . . 6 𝑦𝐴
5 fvopab5.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 nfopab2 5181 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
75, 6nfcxfr 2906 . . . . . 6 𝑦𝐹
8 nfcv 2908 . . . . . 6 𝑦𝑧
94, 7, 8nfbr 5157 . . . . 5 𝑦 𝐴𝐹𝑧
10 nfv 1918 . . . . 5 𝑧 𝐴𝐹𝑦
113, 9, 10cbviotaw 6460 . . . 4 (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
122, 11eqtri 2765 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
13 nfcv 2908 . . . . . . 7 𝑥𝐴
14 nfopab1 5180 . . . . . . . 8 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
155, 14nfcxfr 2906 . . . . . . 7 𝑥𝐹
16 nfcv 2908 . . . . . . 7 𝑥𝑦
1713, 15, 16nfbr 5157 . . . . . 6 𝑥 𝐴𝐹𝑦
18 nfv 1918 . . . . . 6 𝑥𝜓
1917, 18nfbi 1907 . . . . 5 𝑥(𝐴𝐹𝑦𝜓)
20 breq1 5113 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
21 fvopab5.2 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2220, 21bibi12d 346 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝜑) ↔ (𝐴𝐹𝑦𝜓)))
23 df-br 5111 . . . . . 6 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
245eleq2i 2830 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25 opabidw 5486 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2623, 24, 253bitri 297 . . . . 5 (𝑥𝐹𝑦𝜑)
2719, 22, 26vtoclg1f 3527 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑦𝜓))
2827iotabidv 6485 . . 3 (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
2912, 28eqtrid 2789 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑦𝜓))
301, 29syl 17 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3448  cop 4597   class class class wbr 5110  {copab 5172  cio 6451  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-iota 6453  df-fv 6509
This theorem is referenced by:  ajval  29845  adjval  30874
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