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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 5017 | . . 3 | |
2 | fndm 5018 | . . 3 | |
3 | reldm0 4571 | . . . 4 | |
4 | 3 | biimpar 291 | . . 3 |
5 | 1, 2, 4 | syl2anc 403 | . 2 |
6 | fun0 4977 | . . . 4 | |
7 | dm0 4567 | . . . 4 | |
8 | df-fn 4925 | . . . 4 | |
9 | 6, 7, 8 | mpbir2an 883 | . . 3 |
10 | fneq1 5007 | . . 3 | |
11 | 9, 10 | mpbiri 166 | . 2 |
12 | 5, 11 | impbii 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 103 wceq 1284 c0 3251 cdm 4363 wrel 4368 wfun 4916 wfn 4917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-fun 4924 df-fn 4925 |
This theorem is referenced by: mpt0 5046 f0 5100 f00 5101 f1o00 5181 fo00 5182 tpos0 5912 0fz1 9064 |
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