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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 5565 | . . . . . 6 | |
2 | 1 | 3ad2antl2 1155 | . . . . 5 |
3 | fvco3 5565 | . . . . . 6 | |
4 | 3 | 3ad2antl3 1156 | . . . . 5 |
5 | 2, 4 | eqeq12d 2185 | . . . 4 |
6 | simpl1 995 | . . . . 5 | |
7 | ffvelrn 5626 | . . . . . 6 | |
8 | 7 | 3ad2antl2 1155 | . . . . 5 |
9 | ffvelrn 5626 | . . . . . 6 | |
10 | 9 | 3ad2antl3 1156 | . . . . 5 |
11 | f1fveq 5748 | . . . . 5 | |
12 | 6, 8, 10, 11 | syl12anc 1231 | . . . 4 |
13 | 5, 12 | bitrd 187 | . . 3 |
14 | 13 | ralbidva 2466 | . 2 |
15 | f1f 5401 | . . . . . 6 | |
16 | 15 | 3ad2ant1 1013 | . . . . 5 |
17 | ffn 5345 | . . . . 5 | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | simp2 993 | . . . 4 | |
20 | fnfco 5370 | . . . 4 | |
21 | 18, 19, 20 | syl2anc 409 | . . 3 |
22 | simp3 994 | . . . 4 | |
23 | fnfco 5370 | . . . 4 | |
24 | 18, 22, 23 | syl2anc 409 | . . 3 |
25 | eqfnfv 5591 | . . 3 | |
26 | 21, 24, 25 | syl2anc 409 | . 2 |
27 | ffn 5345 | . . . 4 | |
28 | 19, 27 | syl 14 | . . 3 |
29 | ffn 5345 | . . . 4 | |
30 | 22, 29 | syl 14 | . . 3 |
31 | eqfnfv 5591 | . . 3 | |
32 | 28, 30, 31 | syl2anc 409 | . 2 |
33 | 14, 26, 32 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wral 2448 ccom 4613 wfn 5191 wf 5192 wf1 5193 cfv 5196 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fv 5204 |
This theorem is referenced by: mapen 6820 hashfacen 10758 |
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