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Theorem fvopab3ig 5189
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
Hypotheses
Ref Expression
fvopab3ig.1 (x = A → (φψ))
fvopab3ig.2 (y = B → (ψχ))
fvopab3ig.3 (x 𝐶∃*yφ)
fvopab3ig.4 𝐹 = {⟨x, y⟩ ∣ (x 𝐶 φ)}
Assertion
Ref Expression
fvopab3ig ((A 𝐶 B 𝐷) → (χ → (𝐹A) = B))
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 fvopab3ig
StepHypRef Expression
1 eleq1 2097 . . . . . . . 8 (x = A → (x 𝐶A 𝐶))
2 fvopab3ig.1 . . . . . . . 8 (x = A → (φψ))
31, 2anbi12d 442 . . . . . . 7 (x = A → ((x 𝐶 φ) ↔ (A 𝐶 ψ)))
4 fvopab3ig.2 . . . . . . . 8 (y = B → (ψχ))
54anbi2d 437 . . . . . . 7 (y = B → ((A 𝐶 ψ) ↔ (A 𝐶 χ)))
63, 5opelopabg 3996 . . . . . 6 ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)} ↔ (A 𝐶 χ)))
76biimpar 281 . . . . 5 (((A 𝐶 B 𝐷) (A 𝐶 χ)) → ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)})
87exp43 354 . . . 4 (A 𝐶 → (B 𝐷 → (A 𝐶 → (χ → ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)}))))
98pm2.43a 45 . . 3 (A 𝐶 → (B 𝐷 → (χ → ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)})))
109imp 115 . 2 ((A 𝐶 B 𝐷) → (χ → ⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)}))
11 fvopab3ig.4 . . . 4 𝐹 = {⟨x, y⟩ ∣ (x 𝐶 φ)}
1211fveq1i 5122 . . 3 (𝐹A) = ({⟨x, y⟩ ∣ (x 𝐶 φ)}‘A)
13 funopab 4878 . . . . 5 (Fun {⟨x, y⟩ ∣ (x 𝐶 φ)} ↔ x∃*y(x 𝐶 φ))
14 fvopab3ig.3 . . . . . 6 (x 𝐶∃*yφ)
15 moanimv 1972 . . . . . 6 (∃*y(x 𝐶 φ) ↔ (x 𝐶∃*yφ))
1614, 15mpbir 134 . . . . 5 ∃*y(x 𝐶 φ)
1713, 16mpgbir 1339 . . . 4 Fun {⟨x, y⟩ ∣ (x 𝐶 φ)}
18 funopfv 5156 . . . 4 (Fun {⟨x, y⟩ ∣ (x 𝐶 φ)} → (⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)} → ({⟨x, y⟩ ∣ (x 𝐶 φ)}‘A) = B))
1917, 18ax-mp 7 . . 3 (⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)} → ({⟨x, y⟩ ∣ (x 𝐶 φ)}‘A) = B)
2012, 19syl5eq 2081 . 2 (⟨A, B {⟨x, y⟩ ∣ (x 𝐶 φ)} → (𝐹A) = B)
2110, 20syl6 29 1 ((A 𝐶 B 𝐷) → (χ → (𝐹A) = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃*wmo 1898  cop 3370  {copab 3808  Fun wfun 4839  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:  fvmptg  5191  fvopab6  5207  ov6g  5580
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