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Theorem fvopab5 6472
 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 3352 . 2 (𝐴𝑉𝐴 ∈ V)
2 df-fv 6057 . . . 4 (𝐹𝐴) = (℩𝑧𝐴𝐹𝑧)
3 breq2 4808 . . . . 5 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
4 nfcv 2902 . . . . . 6 𝑦𝐴
5 fvopab5.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 nfopab2 4872 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
75, 6nfcxfr 2900 . . . . . 6 𝑦𝐹
8 nfcv 2902 . . . . . 6 𝑦𝑧
94, 7, 8nfbr 4851 . . . . 5 𝑦 𝐴𝐹𝑧
10 nfv 1992 . . . . 5 𝑧 𝐴𝐹𝑦
113, 9, 10cbviota 6017 . . . 4 (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
122, 11eqtri 2782 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
13 nfcv 2902 . . . . . . 7 𝑥𝐴
14 nfopab1 4871 . . . . . . . 8 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
155, 14nfcxfr 2900 . . . . . . 7 𝑥𝐹
16 nfcv 2902 . . . . . . 7 𝑥𝑦
1713, 15, 16nfbr 4851 . . . . . 6 𝑥 𝐴𝐹𝑦
18 nfv 1992 . . . . . 6 𝑥𝜓
1917, 18nfbi 1982 . . . . 5 𝑥(𝐴𝐹𝑦𝜓)
20 breq1 4807 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
21 fvopab5.2 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2220, 21bibi12d 334 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝜑) ↔ (𝐴𝐹𝑦𝜓)))
23 df-br 4805 . . . . . 6 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
245eleq2i 2831 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25 opabid 5132 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2623, 24, 253bitri 286 . . . . 5 (𝑥𝐹𝑦𝜑)
2719, 22, 26vtoclg1f 3405 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑦𝜓))
2827iotabidv 6033 . . 3 (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
2912, 28syl5eq 2806 . 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 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   class class class wbr 4804  {copab 4864  ℩cio 6010  ‘cfv 6049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-iota 6012  df-fv 6057 This theorem is referenced by:  ajval  28026  adjval  29058
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