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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 4639 | . . 3 | |
2 | reldm0 4553 | . . 3 | |
3 | 1, 2 | ax-mp 7 | . 2 |
4 | dmres 4632 | . . . . 5 | |
5 | incom 3129 | . . . . 5 | |
6 | 4, 5 | eqtri 2060 | . . . 4 |
7 | fndm 4998 | . . . . 5 | |
8 | 7 | ineq1d 3137 | . . . 4 |
9 | 6, 8 | syl5eq 2084 | . . 3 |
10 | 9 | eqeq1d 2048 | . 2 |
11 | 3, 10 | syl5rbb 182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wceq 1243 cin 2916 c0 3224 cdm 4345 cres 4347 wrel 4350 wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-dm 4355 df-res 4357 df-fn 4905 |
This theorem is referenced by: fvsnun2 5361 fseq1p1m1 8956 |
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