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Mirrors > Home > ILE Home > Th. List > fnovex | Structured version GIF version |
Description: The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.) |
Ref | Expression |
---|---|
fnovex | ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ A ∈ 𝐶 ∧ B ∈ 𝐷) → (A𝐹B) ∈ V) |
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) |
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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