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Theorem fvopab6 5207
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⟩ ∣ (φ y = B)}
fvopab6.2 (x = A → (φψ))
fvopab6.3 (x = AB = 𝐶)
Assertion
Ref Expression
fvopab6 ((A 𝐷 𝐶 𝑅 ψ) → (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 2560 . . 3 (A 𝐷A V)
2 fvopab6.2 . . . . 5 (x = A → (φψ))
3 fvopab6.3 . . . . . 6 (x = AB = 𝐶)
43eqeq2d 2048 . . . . 5 (x = A → (y = By = 𝐶))
52, 4anbi12d 442 . . . 4 (x = A → ((φ y = B) ↔ (ψ y = 𝐶)))
6 iba 284 . . . . 5 (y = 𝐶 → (ψ ↔ (ψ y = 𝐶)))
76bicomd 129 . . . 4 (y = 𝐶 → ((ψ y = 𝐶) ↔ ψ))
8 moeq 2710 . . . . . 6 ∃*y y = B
98moani 1967 . . . . 5 ∃*y(φ y = B)
109a1i 9 . . . 4 (x V → ∃*y(φ y = B))
11 fvopab6.1 . . . . 5 𝐹 = {⟨x, y⟩ ∣ (φ y = B)}
12 vex 2554 . . . . . . 7 x V
1312biantrur 287 . . . . . 6 ((φ y = B) ↔ (x V (φ y = B)))
1413opabbii 3815 . . . . 5 {⟨x, y⟩ ∣ (φ y = B)} = {⟨x, y⟩ ∣ (x V (φ y = B))}
1511, 14eqtri 2057 . . . 4 𝐹 = {⟨x, y⟩ ∣ (x V (φ y = B))}
165, 7, 10, 15fvopab3ig 5189 . . 3 ((A V 𝐶 𝑅) → (ψ → (𝐹A) = 𝐶))
171, 16sylan 267 . 2 ((A 𝐷 𝐶 𝑅) → (ψ → (𝐹A) = 𝐶))
18173impia 1100 1 ((A 𝐷 𝐶 𝑅 ψ) → (𝐹A) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃*wmo 1898  Vcvv 2551  {copab 3808  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by: (None)
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