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Mirrors > Home > MPE Home > Th. List > fnresdisj | Structured version Visualization version 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 5882 | . . 3 ⊢ Rel (𝐹 ↾ 𝐵) | |
2 | reldm0 5798 | . . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅) |
4 | dmres 5875 | . . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
5 | incom 4178 | . . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2844 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵) |
7 | fndm 6455 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
8 | 7 | ineq1d 4188 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵)) |
9 | 6, 8 | syl5eq 2868 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵)) |
10 | 9 | eqeq1d 2823 | . 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) |
11 | 3, 10 | syl5rbb 286 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∩ cin 3935 ∅c0 4291 dom cdm 5555 ↾ cres 5557 Rel wrel 5560 Fn wfn 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-dm 5565 df-res 5567 df-fn 6358 |
This theorem is referenced by: funressn 6921 fvsnun2 6945 axdc3lem4 9875 fseq1p1m1 12982 hashgval 13694 hashinf 13696 pwssplit1 19831 mplmonmul 20245 wwlksm1edg 27659 eulerpartlemt 31629 poimirlem3 34910 pwssplit4 39709 |
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