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Theorem fvopab3g 5426
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 2162 . . . 4 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3g.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 460 . . 3 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3g.3 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 455 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 4128 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
7 fvopab3g.4 . . . . . 6 (𝑥𝐶 → ∃!𝑦𝜑)
8 fvopab3g.5 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
97, 8fnopab 5183 . . . . 5 𝐹 Fn 𝐶
10 fnopfvb 5395 . . . . 5 ((𝐹 Fn 𝐶𝐴𝐶) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
119, 10mpan 418 . . . 4 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
128eleq2i 2166 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
1311, 12syl6bb 195 . . 3 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
1413adantr 272 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
15 ibar 297 . . 3 (𝐴𝐶 → (𝜒 ↔ (𝐴𝐶𝜒)))
1615adantr 272 . 2 ((𝐴𝐶𝐵𝐷) → (𝜒 ↔ (𝐴𝐶𝜒)))
176, 14, 163bitr4d 219 1 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1299  wcel 1448  ∃!weu 1960  cop 3477  {copab 3928   Fn wfn 5054  cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067
This theorem is referenced by:  recmulnqg  7100
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