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Theorem ovigg 5563
Description: The value of an operation class abstraction. Compare ovig 5564. The condition (x 𝑅 y 𝑆) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovigg.1 ((x = A y = B z = 𝐶) → (φψ))
ovigg.4 ∃*zφ
ovigg.5 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ φ}
Assertion
Ref Expression
ovigg ((A 𝑉 B 𝑊 𝐶 𝑋) → (ψ → (A𝐹B) = 𝐶))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   ψ,x,y,z
Allowed substitution hints:   φ(x,y,z)   𝐹(x,y,z)   𝑉(x,y,z)   𝑊(x,y,z)   𝑋(x,y,z)

Proof of Theorem ovigg
StepHypRef Expression
1 ovigg.1 . . 3 ((x = A y = B z = 𝐶) → (φψ))
21eloprabga 5533 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))
3 df-ov 5458 . . . 4 (A𝐹B) = (𝐹‘⟨A, B⟩)
4 ovigg.5 . . . . 5 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ φ}
54fveq1i 5122 . . . 4 (𝐹‘⟨A, B⟩) = ({⟨⟨x, y⟩, z⟩ ∣ φ}‘⟨A, B⟩)
63, 5eqtri 2057 . . 3 (A𝐹B) = ({⟨⟨x, y⟩, z⟩ ∣ φ}‘⟨A, B⟩)
7 ovigg.4 . . . . 5 ∃*zφ
87funoprab 5543 . . . 4 Fun {⟨⟨x, y⟩, z⟩ ∣ φ}
9 funopfv 5156 . . . 4 (Fun {⟨⟨x, y⟩, z⟩ ∣ φ} → (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} → ({⟨⟨x, y⟩, z⟩ ∣ φ}‘⟨A, B⟩) = 𝐶))
108, 9ax-mp 7 . . 3 (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} → ({⟨⟨x, y⟩, z⟩ ∣ φ}‘⟨A, B⟩) = 𝐶)
116, 10syl5eq 2081 . 2 (⟨⟨A, B⟩, 𝐶 {⟨⟨x, y⟩, z⟩ ∣ φ} → (A𝐹B) = 𝐶)
122, 11syl6bir 153 1 ((A 𝑉 B 𝑊 𝐶 𝑋) → (ψ → (A𝐹B) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃*wmo 1898  cop 3370  Fun wfun 4839  cfv 4845  (class class class)co 5455  {coprab 5456
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-bndl 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  df-ov 5458  df-oprab 5459
This theorem is referenced by:  ovig  5564
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