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Theorem fvopab3ig 6988
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
Hypotheses
Ref Expression
fvopab3ig.1 (𝑥 = 𝐴 → (𝜑𝜓))
fvopab3ig.2 (𝑦 = 𝐵 → (𝜓𝜒))
fvopab3ig.3 (𝑥𝐶 → ∃*𝑦𝜑)
fvopab3ig.4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
Assertion
Ref Expression
fvopab3ig ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab3ig
StepHypRef Expression
1 eleq1 2815 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3ig.1 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 630 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3ig.2 . . . . . . . 8 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 628 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 5531 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
76biimpar 477 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (𝐴𝐶𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
87exp43 436 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝐴𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))))
98pm2.43a 54 . . 3 (𝐴𝐶 → (𝐵𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})))
109imp 406 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
11 fvopab3ig.4 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
1211fveq1i 6886 . . 3 (𝐹𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴)
13 funopab 6577 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐶𝜑))
14 fvopab3ig.3 . . . . . 6 (𝑥𝐶 → ∃*𝑦𝜑)
15 moanimv 2609 . . . . . 6 (∃*𝑦(𝑥𝐶𝜑) ↔ (𝑥𝐶 → ∃*𝑦𝜑))
1614, 15mpbir 230 . . . . 5 ∃*𝑦(𝑥𝐶𝜑)
1713, 16mpgbir 1793 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
18 funopfv 6937 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴) = 𝐵))
1917, 18ax-mp 5 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴) = 𝐵)
2012, 19eqtrid 2778 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → (𝐹𝐴) = 𝐵)
2110, 20syl6 35 1 ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  ∃*wmo 2526  cop 4629  {copab 5203  Fun wfun 6531  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by:  fvmptg  6990  fvopab6  7025  ov6g  7568
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