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Mirrors > Home > ILE Home > Th. List > fnovex | 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 5839 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | opelxp 4628 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
3 | funfvex 5497 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) | |
4 | 3 | funfni 5282 | . . . 4 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) |
5 | 2, 4 | sylan2br 286 | . . 3 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) |
6 | 5 | 3impb 1188 | . 2 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) |
7 | 1, 6 | eqeltrid 2251 | 1 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 ∈ wcel 2135 Vcvv 2721 〈cop 3573 × cxp 4596 Fn wfn 5177 ‘cfv 5182 (class class class)co 5836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-ov 5839 |
This theorem is referenced by: ovelrn 5981 mapsnen 6768 map1 6769 mapen 6803 mapdom1g 6804 mapxpen 6805 xpmapenlem 6806 fzen 9968 hashfacen 10735 omctfn 12313 topnfn 12497 topnvalg 12504 restbasg 12709 tgrest 12710 restco 12715 lmfval 12733 cnfval 12735 cnpfval 12736 cnpval 12739 txrest 12817 ismet 12885 isxmet 12886 xmetunirn 12899 |
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