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Theorem fvopab3g 6539
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 2847 . . . 4 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3g.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 624 . . 3 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3g.3 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 622 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 5232 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
7 fvopab3g.4 . . . . . 6 (𝑥𝐶 → ∃!𝑦𝜑)
8 fvopab3g.5 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
97, 8fnopab 6266 . . . . 5 𝐹 Fn 𝐶
10 fnopfvb 6498 . . . . 5 ((𝐹 Fn 𝐶𝐴𝐶) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
119, 10mpan 680 . . . 4 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
128eleq2i 2851 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
1311, 12syl6bb 279 . . 3 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
1413adantr 474 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
15 ibar 524 . . 3 (𝐴𝐶 → (𝜒 ↔ (𝐴𝐶𝜒)))
1615adantr 474 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 ↔ (𝐴𝐶𝜒)))
176, 14, 163bitr4d 303 1 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  ∃!weu 2586  cop 4404  {copab 4950   Fn wfn 6132  cfv 6137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fn 6140  df-fv 6145
This theorem is referenced by:  recmulnq  10123
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