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Theorem ovidig 5560
Description: The value of an operation class abstraction. Compare ovidi 5561. The condition (x 𝑅 y 𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovidig.1 ∃*zφ
ovidig.2 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ φ}
Assertion
Ref Expression
ovidig (φ → (x𝐹y) = z)
Distinct variable group:   x,y,z
Allowed substitution hints:   φ(x,y,z)   𝐹(x,y,z)

Proof of Theorem ovidig
StepHypRef Expression
1 df-ov 5458 . 2 (x𝐹y) = (𝐹‘⟨x, y⟩)
2 ovidig.1 . . . . 5 ∃*zφ
32funoprab 5543 . . . 4 Fun {⟨⟨x, y⟩, z⟩ ∣ φ}
4 ovidig.2 . . . . 5 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ φ}
54funeqi 4865 . . . 4 (Fun 𝐹 ↔ Fun {⟨⟨x, y⟩, z⟩ ∣ φ})
63, 5mpbir 134 . . 3 Fun 𝐹
7 oprabid 5480 . . . . 5 (⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ φ)
87biimpri 124 . . . 4 (φ → ⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ})
98, 4syl6eleqr 2128 . . 3 (φ → ⟨⟨x, y⟩, z 𝐹)
10 funopfv 5156 . . 3 (Fun 𝐹 → (⟨⟨x, y⟩, z 𝐹 → (𝐹‘⟨x, y⟩) = z))
116, 9, 10mpsyl 59 . 2 (φ → (𝐹‘⟨x, y⟩) = z)
121, 11syl5eq 2081 1 (φ → (x𝐹y) = z)
Colors of variables: wff set class
Syntax hints:  wi 4   = 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-in1 544  ax-in2 545  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  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  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:  ovidi  5561
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