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Theorem fnovex 5460
Description: The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
Assertion
Ref Expression
fnovex ((𝐹 Fn (𝐶 × 𝐷) A 𝐶 B 𝐷) → (A𝐹B) V)

Proof of Theorem fnovex
StepHypRef Expression
1 df-ov 5437 . 2 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 opelxp 4299 . . . 4 (⟨A, B (𝐶 × 𝐷) ↔ (A 𝐶 B 𝐷))
3 funfvex 5115 . . . . 5 ((Fun 𝐹 A, B dom 𝐹) → (𝐹‘⟨A, B⟩) V)
43funfni 4923 . . . 4 ((𝐹 Fn (𝐶 × 𝐷) A, B (𝐶 × 𝐷)) → (𝐹‘⟨A, B⟩) V)
52, 4sylan2br 272 . . 3 ((𝐹 Fn (𝐶 × 𝐷) (A 𝐶 B 𝐷)) → (𝐹‘⟨A, B⟩) V)
653impb 1086 . 2 ((𝐹 Fn (𝐶 × 𝐷) A 𝐶 B 𝐷) → (𝐹‘⟨A, B⟩) V)
71, 6syl5eqel 2107 1 ((𝐹 Fn (𝐶 × 𝐷) A 𝐶 B 𝐷) → (A𝐹B) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   wcel 1375  Vcvv 2534  cop 3352   × cxp 4268   Fn wfn 4822  cfv 4827  (class class class)co 5434
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-id 4003  df-xp 4276  df-cnv 4278  df-co 4279  df-dm 4280  df-iota 4792  df-fun 4829  df-fn 4830  df-fv 4835  df-ov 5437
This theorem is referenced by:  ovelrn  5570  fnofval  5641
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