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Theorem fvopab3g 6766
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 2903 . . . 4 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3g.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 632 . . 3 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3g.3 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 630 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 5428 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
7 fvopab3g.4 . . . . . 6 (𝑥𝐶 → ∃!𝑦𝜑)
8 fvopab3g.5 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
97, 8fnopab 6489 . . . . 5 𝐹 Fn 𝐶
10 fnopfvb 6722 . . . . 5 ((𝐹 Fn 𝐶𝐴𝐶) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
119, 10mpan 688 . . . 4 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
128eleq2i 2907 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
1311, 12syl6bb 289 . . 3 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
1413adantr 483 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
15 ibar 531 . . 3 (𝐴𝐶 → (𝜒 ↔ (𝐴𝐶𝜒)))
1615adantr 483 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 ↔ (𝐴𝐶𝜒)))
176, 14, 163bitr4d 313 1 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  ∃!weu 2652  cop 4576  {copab 5131   Fn wfn 6353  cfv 6358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366
This theorem is referenced by:  recmulnq  10389
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