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Theorem fvopab3ig 6235
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 2686 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3ig.1 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 746 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3ig.2 . . . . . . . 8 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 739 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 4953 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
76biimpar 502 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (𝐴𝐶𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
87exp43 639 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝐴𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))))
98pm2.43a 54 . . 3 (𝐴𝐶 → (𝐵𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})))
109imp 445 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
11 fvopab3ig.4 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
1211fveq1i 6149 . . 3 (𝐹𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}‘𝐴)
13 funopab 5881 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐶𝜑))
14 fvopab3ig.3 . . . . . 6 (𝑥𝐶 → ∃*𝑦𝜑)
15 moanimv 2530 . . . . . 6 (∃*𝑦(𝑥𝐶𝜑) ↔ (𝑥𝐶 → ∃*𝑦𝜑))
1614, 15mpbir 221 . . . . 5 ∃*𝑦(𝑥𝐶𝜑)
1713, 16mpgbir 1723 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
18 funopfv 6192 . . . 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 2667 . 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 196  wa 384   = wceq 1480  wcel 1987  ∃*wmo 2470  cop 4154  {copab 4672  Fun wfun 5841  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  fvmptg  6237  fvopab6  6266  ov6g  6751
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