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Theorem fvopab3ig 7005
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 2814 . . . . . . . 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 5544 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
76biimpar 476 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (𝐴𝐶𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
87exp43 435 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝐴𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))))
98pm2.43a 54 . . 3 (𝐴𝐶 → (𝐵𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})))
109imp 405 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
11 fvopab3ig.4 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
1211fveq1i 6902 . . 3 (𝐹𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴)
13 funopab 6594 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐶𝜑))
14 fvopab3ig.3 . . . . . 6 (𝑥𝐶 → ∃*𝑦𝜑)
15 moanimv 2608 . . . . . 6 (∃*𝑦(𝑥𝐶𝜑) ↔ (𝑥𝐶 → ∃*𝑦𝜑))
1614, 15mpbir 230 . . . . 5 ∃*𝑦(𝑥𝐶𝜑)
1713, 16mpgbir 1794 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
18 funopfv 6953 . . . 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 394   = wceq 1534  wcel 2099  ∃*wmo 2527  cop 4639  {copab 5215  Fun wfun 6548  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562
This theorem is referenced by:  fvmptg  7007  fvopab6  7043  ov6g  7590
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