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Theorem fvopab6 5291
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 2583 . . 3 (𝐴𝐷𝐴 ∈ V)
2 fvopab6.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
3 fvopab6.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
43eqeq2d 2067 . . . . 5 (𝑥 = 𝐴 → (𝑦 = 𝐵𝑦 = 𝐶))
52, 4anbi12d 450 . . . 4 (𝑥 = 𝐴 → ((𝜑𝑦 = 𝐵) ↔ (𝜓𝑦 = 𝐶)))
6 iba 288 . . . . 5 (𝑦 = 𝐶 → (𝜓 ↔ (𝜓𝑦 = 𝐶)))
76bicomd 133 . . . 4 (𝑦 = 𝐶 → ((𝜓𝑦 = 𝐶) ↔ 𝜓))
8 moeq 2738 . . . . . 6 ∃*𝑦 𝑦 = 𝐵
98moani 1986 . . . . 5 ∃*𝑦(𝜑𝑦 = 𝐵)
109a1i 9 . . . 4 (𝑥 ∈ V → ∃*𝑦(𝜑𝑦 = 𝐵))
11 fvopab6.1 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}
12 vex 2577 . . . . . . 7 𝑥 ∈ V
1312biantrur 291 . . . . . 6 ((𝜑𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵)))
1413opabbii 3851 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵))}
1511, 14eqtri 2076 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑𝑦 = 𝐵))}
165, 7, 10, 15fvopab3ig 5273 . . 3 ((𝐴 ∈ V ∧ 𝐶𝑅) → (𝜓 → (𝐹𝐴) = 𝐶))
171, 16sylan 271 . 2 ((𝐴𝐷𝐶𝑅) → (𝜓 → (𝐹𝐴) = 𝐶))
18173impia 1112 1 ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  ∃*wmo 1917  Vcvv 2574  {copab 3844  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937
This theorem is referenced by: (None)
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