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Mirrors > Home > ILE Home > Th. List > fn0 | Unicode version |
Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 4997 | . . 3 | |
2 | fndm 4998 | . . 3 | |
3 | reldm0 4553 | . . . 4 | |
4 | 3 | biimpar 281 | . . 3 |
5 | 1, 2, 4 | syl2anc 391 | . 2 |
6 | fun0 4957 | . . . 4 | |
7 | dm0 4549 | . . . 4 | |
8 | df-fn 4905 | . . . 4 | |
9 | 6, 7, 8 | mpbir2an 849 | . . 3 |
10 | fneq1 4987 | . . 3 | |
11 | 9, 10 | mpbiri 157 | . 2 |
12 | 5, 11 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 98 wceq 1243 c0 3224 cdm 4345 wrel 4350 wfun 4896 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-nul 3883 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-eu 1903 df-mo 1904 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-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 df-fn 4905 |
This theorem is referenced by: mpt0 5026 f0 5080 f00 5081 f1o00 5161 fo00 5162 tpos0 5889 0fz1 8909 |
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