Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5745 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | opelxp 4539 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
3 | funfvex 5406 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 〈𝐴, 𝐵〉 ∈ dom 𝐹) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) | |
4 | 3 | funfni 5193 | . . . 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 1162 | . 2 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) |
7 | 1, 6 | eqeltrid 2204 | 1 ⊢ ((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 947 ∈ wcel 1465 Vcvv 2660 〈cop 3500 × cxp 4507 Fn wfn 5088 ‘cfv 5093 (class class class)co 5742 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-ov 5745 |
This theorem is referenced by: ovelrn 5887 mapsnen 6673 map1 6674 mapen 6708 mapdom1g 6709 mapxpen 6710 xpmapenlem 6711 fzen 9791 hashfacen 10547 topnfn 12052 topnvalg 12059 restbasg 12264 tgrest 12265 restco 12270 lmfval 12288 cnfval 12290 cnpfval 12291 cnpval 12294 txrest 12372 ismet 12440 isxmet 12441 xmetunirn 12454 |
Copyright terms: Public domain | W3C validator |