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

Step	Hyp	Ref	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)
4	3	funfni 4923	. . . 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 1086	. 2 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ A ∈ 𝐶 ∧ B ∈ 𝐷) → (𝐹‘⟨A, B⟩) ∈ V)
7	1, 6	syl5eqel 2107	1 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ A ∈ 𝐶 ∧ B ∈ 𝐷) → (A𝐹B) ∈ V)

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