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Mirrors > Home > ILE Home > Th. List > fnresdisj | GIF version |
Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.) |
Ref | Expression |
---|---|
fnresdisj | ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4917 | . . 3 ⊢ Rel (𝐹 ↾ 𝐵) | |
2 | reldm0 4827 | . . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅) |
4 | dmres 4910 | . . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
5 | incom 3319 | . . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2191 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵) |
7 | fndm 5295 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
8 | 7 | ineq1d 3327 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵)) |
9 | 6, 8 | eqtrid 2215 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵)) |
10 | 9 | eqeq1d 2179 | . 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) |
11 | 3, 10 | bitr2id 192 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 ∩ cin 3120 ∅c0 3414 dom cdm 4609 ↾ cres 4611 Rel wrel 4614 Fn wfn 5191 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-dm 4619 df-res 4621 df-fn 5199 |
This theorem is referenced by: fvsnun2 5692 fseq1p1m1 10039 |
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