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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 5492 | . . . . . 6 | |
2 | 1 | 3ad2antl2 1144 | . . . . 5 |
3 | fvco3 5492 | . . . . . 6 | |
4 | 3 | 3ad2antl3 1145 | . . . . 5 |
5 | 2, 4 | eqeq12d 2154 | . . . 4 |
6 | simpl1 984 | . . . . 5 | |
7 | ffvelrn 5553 | . . . . . 6 | |
8 | 7 | 3ad2antl2 1144 | . . . . 5 |
9 | ffvelrn 5553 | . . . . . 6 | |
10 | 9 | 3ad2antl3 1145 | . . . . 5 |
11 | f1fveq 5673 | . . . . 5 | |
12 | 6, 8, 10, 11 | syl12anc 1214 | . . . 4 |
13 | 5, 12 | bitrd 187 | . . 3 |
14 | 13 | ralbidva 2433 | . 2 |
15 | f1f 5328 | . . . . . 6 | |
16 | 15 | 3ad2ant1 1002 | . . . . 5 |
17 | ffn 5272 | . . . . 5 | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | simp2 982 | . . . 4 | |
20 | fnfco 5297 | . . . 4 | |
21 | 18, 19, 20 | syl2anc 408 | . . 3 |
22 | simp3 983 | . . . 4 | |
23 | fnfco 5297 | . . . 4 | |
24 | 18, 22, 23 | syl2anc 408 | . . 3 |
25 | eqfnfv 5518 | . . 3 | |
26 | 21, 24, 25 | syl2anc 408 | . 2 |
27 | ffn 5272 | . . . 4 | |
28 | 19, 27 | syl 14 | . . 3 |
29 | ffn 5272 | . . . 4 | |
30 | 22, 29 | syl 14 | . . 3 |
31 | eqfnfv 5518 | . . 3 | |
32 | 28, 30, 31 | syl2anc 408 | . 2 |
33 | 14, 26, 32 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 ccom 4543 wfn 5118 wf 5119 wf1 5120 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fv 5131 |
This theorem is referenced by: mapen 6740 hashfacen 10579 |
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