Theorem fnovex 5431

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

Step	Hyp	Ref	Expression
1		df-ov 5408	. 2 ⊢ (A𝐹B) = (𝐹‘⟨A, B⟩)
2		opelxp 4267	. . . 4 ⊢ (⟨A, B⟩ ∈ (𝐶 × 𝐷) ↔ (A ∈ 𝐶 ∧ B ∈ 𝐷))
3		funfvex 5084	. . . . 5 ⊢ ((Fun 𝐹 ∧ ⟨A, B⟩ ∈ dom 𝐹) → (𝐹‘⟨A, B⟩) ∈ V)
4	3	funfni 4892	. . . 4 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ ⟨A, B⟩ ∈ (𝐶 × 𝐷)) → (𝐹‘⟨A, B⟩) ∈ V)
5	2, 4	sylan2br 272	. . 3 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ (A ∈ 𝐶 ∧ B ∈ 𝐷)) → (𝐹‘⟨A, B⟩) ∈ V)
6	5	3impb 1088	. 2 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ A ∈ 𝐶 ∧ B ∈ 𝐷) → (𝐹‘⟨A, B⟩) ∈ V)
7	1, 6	syl5eqel 2106	1 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ A ∈ 𝐶 ∧ B ∈ 𝐷) → (A𝐹B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 875   ∈ wcel 1375  Vcvv 2533  ⟨cop 3330   × cxp 4236   Fn wfn 4791  ‘cfv 4796  (class class class)co 5405

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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-id 3983  df-xp 4244  df-cnv 4246  df-co 4247  df-dm 4248  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804  df-ov 5408

This theorem is referenced by:  ovelrn  5541  fnofval  5610