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Theorem fnovex 5481
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 5458 . 2 (A𝐹B) = (𝐹‘⟨A, B⟩)
2 opelxp 4317 . . . 4 (⟨A, B (𝐶 × 𝐷) ↔ (A 𝐶 B 𝐷))
3 funfvex 5135 . . . . 5 ((Fun 𝐹 A, B dom 𝐹) → (𝐹‘⟨A, B⟩) V)
43funfni 4942 . . . 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 1099 . 2 ((𝐹 Fn (𝐶 × 𝐷) A 𝐶 B 𝐷) → (𝐹‘⟨A, B⟩) V)
71, 6syl5eqel 2121 1 ((𝐹 Fn (𝐶 × 𝐷) A 𝐶 B 𝐷) → (A𝐹B) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   wcel 1390  Vcvv 2551  cop 3370   × cxp 4286   Fn wfn 4840  cfv 4845  (class class class)co 5455
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-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458
This theorem is referenced by:  ovelrn  5591  fnofval  5663  fzen  8677
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