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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 5964 | . . 3 ⊢ Rel (𝐹 ↾ 𝐵) | |
2 | reldm0 5881 | . . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅) |
4 | dmres 5957 | . . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
5 | incom 4159 | . . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2764 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵) |
7 | fndm 6602 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
8 | 7 | ineq1d 4169 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵)) |
9 | 6, 8 | eqtrid 2788 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵)) |
10 | 9 | eqeq1d 2738 | . 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) |
11 | 3, 10 | bitr2id 283 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∩ cin 3907 ∅c0 4280 dom cdm 5631 ↾ cres 5633 Rel wrel 5636 Fn wfn 6488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-dm 5641 df-res 5643 df-fn 6496 |
This theorem is referenced by: funressn 7101 fvsnun2 7125 dif1enlem 9096 dif1enlemOLD 9097 axdc3lem4 10385 fseq1p1m1 13507 hashgval 14225 hashinf 14227 pwssplit1 20505 mplmonmul 21421 wwlksm1edg 28712 eulerpartlemt 32840 poimirlem3 36048 pwssplit4 41354 |
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