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Theorem fvopab3ig 6173
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 2675 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3ig.1 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 742 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3ig.2 . . . . . . . 8 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 735 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 4908 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
76biimpar 500 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (𝐴𝐶𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
87exp43 637 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝐴𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))))
98pm2.43a 51 . . 3 (𝐴𝐶 → (𝐵𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})))
109imp 443 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
11 fvopab3ig.4 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
1211fveq1i 6089 . . 3 (𝐹𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴)
13 funopab 5823 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐶𝜑))
14 fvopab3ig.3 . . . . . 6 (𝑥𝐶 → ∃*𝑦𝜑)
15 moanimv 2518 . . . . . 6 (∃*𝑦(𝑥𝐶𝜑) ↔ (𝑥𝐶 → ∃*𝑦𝜑))
1614, 15mpbir 219 . . . . 5 ∃*𝑦(𝑥𝐶𝜑)
1713, 16mpgbir 1716 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
18 funopfv 6130 . . . 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, 19syl5eq 2655 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} → (𝐹𝐴) = 𝐵)
2110, 20syl6 34 1 ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  ∃*wmo 2458  cop 4130  {copab 4636  Fun wfun 5784  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  fvmptg  6174  fvopab6  6203  ov6g  6674
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