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Theorem fvopab3g 5412
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 2157 . . . 4 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 fvopab3g.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2anbi12d 458 . . 3 (𝑥 = 𝐴 → ((𝑥𝐶𝜑) ↔ (𝐴𝐶𝜓)))
4 fvopab3g.3 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
54anbi2d 453 . . 3 (𝑦 = 𝐵 → ((𝐴𝐶𝜓) ↔ (𝐴𝐶𝜒)))
63, 5opelopabg 4119 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)} ↔ (𝐴𝐶𝜒)))
7 fvopab3g.4 . . . . . 6 (𝑥𝐶 → ∃!𝑦𝜑)
8 fvopab3g.5 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}
97, 8fnopab 5172 . . . . 5 𝐹 Fn 𝐶
10 fnopfvb 5381 . . . . 5 ((𝐹 Fn 𝐶𝐴𝐶) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
119, 10mpan 416 . . . 4 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
128eleq2i 2161 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)})
1311, 12syl6bb 195 . . 3 (𝐴𝐶 → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
1413adantr 271 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}))
15 ibar 296 . . 3 (𝐴𝐶 → (𝜒 ↔ (𝐴𝐶𝜒)))
1615adantr 271 . 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 1296  wcel 1445  ∃!weu 1955  cop 3469  {copab 3920   Fn wfn 5044  cfv 5049
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057
This theorem is referenced by:  recmulnqg  7047
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