Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cocan1 | Unicode version |
Description: An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
cocan1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvco3 5244 | . . . . . 6 | |
2 | 1 | 3ad2antl2 1067 | . . . . 5 |
3 | fvco3 5244 | . . . . . 6 | |
4 | 3 | 3ad2antl3 1068 | . . . . 5 |
5 | 2, 4 | eqeq12d 2054 | . . . 4 |
6 | simpl1 907 | . . . . 5 | |
7 | ffvelrn 5300 | . . . . . 6 | |
8 | 7 | 3ad2antl2 1067 | . . . . 5 |
9 | ffvelrn 5300 | . . . . . 6 | |
10 | 9 | 3ad2antl3 1068 | . . . . 5 |
11 | f1fveq 5411 | . . . . 5 | |
12 | 6, 8, 10, 11 | syl12anc 1133 | . . . 4 |
13 | 5, 12 | bitrd 177 | . . 3 |
14 | 13 | ralbidva 2322 | . 2 |
15 | f1f 5092 | . . . . . 6 | |
16 | 15 | 3ad2ant1 925 | . . . . 5 |
17 | ffn 5046 | . . . . 5 | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | simp2 905 | . . . 4 | |
20 | fnfco 5065 | . . . 4 | |
21 | 18, 19, 20 | syl2anc 391 | . . 3 |
22 | simp3 906 | . . . 4 | |
23 | fnfco 5065 | . . . 4 | |
24 | 18, 22, 23 | syl2anc 391 | . . 3 |
25 | eqfnfv 5265 | . . 3 | |
26 | 21, 24, 25 | syl2anc 391 | . 2 |
27 | ffn 5046 | . . . 4 | |
28 | 19, 27 | syl 14 | . . 3 |
29 | ffn 5046 | . . . 4 | |
30 | 22, 29 | syl 14 | . . 3 |
31 | eqfnfv 5265 | . . 3 | |
32 | 28, 30, 31 | syl2anc 391 | . 2 |
33 | 14, 26, 32 | 3bitr4d 209 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 wral 2306 ccom 4349 wfn 4897 wf 4898 wf1 4899 cfv 4902 |
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-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-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-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fv 4910 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |