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Mirrors > Home > ILE Home > Th. List > fnresdisj | Unicode version |
Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.) |
Ref | Expression |
---|---|
fnresdisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4817 | . . 3 | |
2 | reldm0 4727 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | dmres 4810 | . . . . 5 | |
5 | incom 3238 | . . . . 5 | |
6 | 4, 5 | eqtri 2138 | . . . 4 |
7 | fndm 5192 | . . . . 5 | |
8 | 7 | ineq1d 3246 | . . . 4 |
9 | 6, 8 | syl5eq 2162 | . . 3 |
10 | 9 | eqeq1d 2126 | . 2 |
11 | 3, 10 | syl5rbb 192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1316 cin 3040 c0 3333 cdm 4509 cres 4511 wrel 4514 wfn 5088 |
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 588 ax-in2 589 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-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-dm 4519 df-res 4521 df-fn 5096 |
This theorem is referenced by: fvsnun2 5586 fseq1p1m1 9842 |
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