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Theorem fvopab6 6267
Description: Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvopab6.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}
fvopab6.2 (𝑥 = 𝐴 → (𝜑𝜓))
fvopab6.3 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
fvopab6 ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜓,𝑥,𝑦   𝑦,𝐵   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab6
StepHypRef Expression
1 elex 3203 . . 3 (𝐴𝐷𝐴 ∈ V)
2 fvopab6.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
3 fvopab6.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
43eqeq2d 2636 . . . . 5 (𝑥 = 𝐴 → (𝑦 = 𝐵𝑦 = 𝐶))
52, 4anbi12d 746 . . . 4 (𝑥 = 𝐴 → ((𝜑𝑦 = 𝐵) ↔ (𝜓𝑦 = 𝐶)))
6 iba 524 . . . . 5 (𝑦 = 𝐶 → (𝜓 ↔ (𝜓𝑦 = 𝐶)))
76bicomd 213 . . . 4 (𝑦 = 𝐶 → ((𝜓𝑦 = 𝐶) ↔ 𝜓))
8 moeq 3369 . . . . . 6 ∃*𝑦 𝑦 = 𝐵
98moani 2529 . . . . 5 ∃*𝑦(𝜑𝑦 = 𝐵)
109a1i 11 . . . 4 (𝑥 ∈ V → ∃*𝑦(𝜑𝑦 = 𝐵))
11 fvopab6.1 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}
12 vex 3194 . . . . . . 7 𝑥 ∈ V
1312biantrur 527 . . . . . 6 ((𝜑𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵)))
1413opabbii 4684 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵))}
1511, 14eqtri 2648 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵))}
165, 7, 10, 15fvopab3ig 6236 . . 3 ((𝐴 ∈ V ∧ 𝐶𝑅) → (𝜓 → (𝐹𝐴) = 𝐶))
171, 16sylan 488 . 2 ((𝐴𝐷𝐶𝑅) → (𝜓 → (𝐹𝐴) = 𝐶))
18173impia 1258 1 ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  ∃*wmo 2475  Vcvv 3191  {copab 4677  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858
This theorem is referenced by: (None)
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