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Theorem fvopab5 6531
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 3406 . 2 (𝐴𝑉𝐴 ∈ V)
2 df-fv 6109 . . . 4 (𝐹𝐴) = (℩𝑧𝐴𝐹𝑧)
3 breq2 4848 . . . . 5 (𝑧 = 𝑦 → (𝐴𝐹𝑧𝐴𝐹𝑦))
4 nfcv 2948 . . . . . 6 𝑦𝐴
5 fvopab5.1 . . . . . . 7 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 nfopab2 4914 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
75, 6nfcxfr 2946 . . . . . 6 𝑦𝐹
8 nfcv 2948 . . . . . 6 𝑦𝑧
94, 7, 8nfbr 4891 . . . . 5 𝑦 𝐴𝐹𝑧
10 nfv 2005 . . . . 5 𝑧 𝐴𝐹𝑦
113, 9, 10cbviota 6069 . . . 4 (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
122, 11eqtri 2828 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
13 nfcv 2948 . . . . . . 7 𝑥𝐴
14 nfopab1 4913 . . . . . . . 8 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
155, 14nfcxfr 2946 . . . . . . 7 𝑥𝐹
16 nfcv 2948 . . . . . . 7 𝑥𝑦
1713, 15, 16nfbr 4891 . . . . . 6 𝑥 𝐴𝐹𝑦
18 nfv 2005 . . . . . 6 𝑥𝜓
1917, 18nfbi 1995 . . . . 5 𝑥(𝐴𝐹𝑦𝜓)
20 breq1 4847 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
21 fvopab5.2 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2220, 21bibi12d 336 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝜑) ↔ (𝐴𝐹𝑦𝜓)))
23 df-br 4845 . . . . . 6 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
245eleq2i 2877 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25 opabid 5177 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
2623, 24, 253bitri 288 . . . . 5 (𝑥𝐹𝑦𝜑)
2719, 22, 26vtoclg1f 3458 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑦𝜓))
2827iotabidv 6085 . . 3 (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
2912, 28syl5eq 2852 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑦𝜓))
301, 29syl 17 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wcel 2156  Vcvv 3391  cop 4376   class class class wbr 4844  {copab 4906  cio 6062  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-iota 6064  df-fv 6109
This theorem is referenced by:  ajval  28045  adjval  29077
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