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Theorem fvopab6 6792
 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 3428 . . 3 (𝐴𝐷𝐴 ∈ V)
2 fvopab6.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
3 fvopab6.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
43eqeq2d 2769 . . . . 5 (𝑥 = 𝐴 → (𝑦 = 𝐵𝑦 = 𝐶))
52, 4anbi12d 633 . . . 4 (𝑥 = 𝐴 → ((𝜑𝑦 = 𝐵) ↔ (𝜓𝑦 = 𝐶)))
6 iba 531 . . . . 5 (𝑦 = 𝐶 → (𝜓 ↔ (𝜓𝑦 = 𝐶)))
76bicomd 226 . . . 4 (𝑦 = 𝐶 → ((𝜓𝑦 = 𝐶) ↔ 𝜓))
8 moeq 3621 . . . . . 6 ∃*𝑦 𝑦 = 𝐵
98moani 2571 . . . . 5 ∃*𝑦(𝜑𝑦 = 𝐵)
109a1i 11 . . . 4 (𝑥 ∈ V → ∃*𝑦(𝜑𝑦 = 𝐵))
11 fvopab6.1 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}
12 vex 3413 . . . . . . 7 𝑥 ∈ V
1312biantrur 534 . . . . . 6 ((𝜑𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵)))
1413opabbii 5099 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵))}
1511, 14eqtri 2781 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵))}
165, 7, 10, 15fvopab3ig 6755 . . 3 ((𝐴 ∈ V ∧ 𝐶𝑅) → (𝜓 → (𝐹𝐴) = 𝐶))
171, 16sylan 583 . 2 ((𝐴𝐷𝐶𝑅) → (𝜓 → (𝐹𝐴) = 𝐶))
18173impia 1114 1 ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃*wmo 2555  Vcvv 3409  {copab 5094  ‘cfv 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343 This theorem is referenced by: (None)
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