Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5245 | . . . . 5 | |
2 | frn 5251 | . . . . . . 7 | |
3 | ss0 3373 | . . . . . . 7 | |
4 | 2, 3 | syl 14 | . . . . . 6 |
5 | dm0rn0 4726 | . . . . . 6 | |
6 | 4, 5 | sylibr 133 | . . . . 5 |
7 | df-fn 5096 | . . . . 5 | |
8 | 1, 6, 7 | sylanbrc 413 | . . . 4 |
9 | fn0 5212 | . . . 4 | |
10 | 8, 9 | sylib 121 | . . 3 |
11 | fdm 5248 | . . . 4 | |
12 | 11, 6 | eqtr3d 2152 | . . 3 |
13 | 10, 12 | jca 304 | . 2 |
14 | f0 5283 | . . 3 | |
15 | feq1 5225 | . . . 4 | |
16 | feq2 5226 | . . . 4 | |
17 | 15, 16 | sylan9bb 457 | . . 3 |
18 | 14, 17 | mpbiri 167 | . 2 |
19 | 13, 18 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wss 3041 c0 3333 cdm 4509 crn 4510 wfun 5087 wfn 5088 wf 5089 |
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-nul 4024 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-eu 1980 df-mo 1981 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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 |
This theorem is referenced by: dom0 6700 |
Copyright terms: Public domain | W3C validator |