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Theorem fvopab3g 6994
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
fvopab3g.2 (𝑥 = 𝐴 → (𝜑𝜓))
fvopab3g.3 (𝑦 = 𝐵 → (𝜓𝜒))
fvopab3g.4 (𝑥𝐶 → ∃!𝑦𝜑)
fvopab3g.5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
Assertion
Ref Expression
fvopab3g ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
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 fvopab3g
StepHypRef Expression
1 eleq1 2817 . . . 4 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3g.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 631 . . 3 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3g.3 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 629 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 5534 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
7 fvopab3g.4 . . . . . 6 (𝑥𝐶 → ∃!𝑦𝜑)
8 fvopab3g.5 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
97, 8fnopab 6687 . . . . 5 𝐹 Fn 𝐶
10 fnopfvb 6945 . . . . 5 ((𝐹 Fn 𝐶𝐴𝐶) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
119, 10mpan 689 . . . 4 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
128eleq2i 2821 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
1311, 12bitrdi 287 . . 3 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
1413adantr 480 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
15 ibar 528 . . 3 (𝐴𝐶 → (𝜒 ↔ (𝐴𝐶𝜒)))
1615adantr 480 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 ↔ (𝐴𝐶𝜒)))
176, 14, 163bitr4d 311 1 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  ∃!weu 2558  cop 4630  {copab 5204   Fn wfn 6537  cfv 6542
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 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  recmulnq  10981
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