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Mirrors > Home > ILE Home > Th. List > f00 | Unicode version |
Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
f00 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 5048 | . . . . 5 | |
2 | frn 5052 | . . . . . . 7 | |
3 | ss0 3257 | . . . . . . 7 | |
4 | 2, 3 | syl 14 | . . . . . 6 |
5 | dm0rn0 4552 | . . . . . 6 | |
6 | 4, 5 | sylibr 137 | . . . . 5 |
7 | df-fn 4905 | . . . . 5 | |
8 | 1, 6, 7 | sylanbrc 394 | . . . 4 |
9 | fn0 5018 | . . . 4 | |
10 | 8, 9 | sylib 127 | . . 3 |
11 | fdm 5050 | . . . 4 | |
12 | 11, 6 | eqtr3d 2074 | . . 3 |
13 | 10, 12 | jca 290 | . 2 |
14 | f0 5080 | . . 3 | |
15 | feq1 5030 | . . . 4 | |
16 | feq2 5031 | . . . 4 | |
17 | 15, 16 | sylan9bb 435 | . . 3 |
18 | 14, 17 | mpbiri 157 | . 2 |
19 | 13, 18 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wss 2917 c0 3224 cdm 4345 crn 4346 wfun 4896 wfn 4897 wf 4898 |
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-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: (None) |
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