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Mirrors > Home > ILE Home > Th. List > fcof1 | Unicode version |
Description: An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
fcof1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . 2 | |
2 | simprr 498 | . . . . . . . 8 | |
3 | 2 | fveq2d 5202 | . . . . . . 7 |
4 | simpll 495 | . . . . . . . 8 | |
5 | simprll 503 | . . . . . . . 8 | |
6 | fvco3 5265 | . . . . . . . 8 | |
7 | 4, 5, 6 | syl2anc 403 | . . . . . . 7 |
8 | simprlr 504 | . . . . . . . 8 | |
9 | fvco3 5265 | . . . . . . . 8 | |
10 | 4, 8, 9 | syl2anc 403 | . . . . . . 7 |
11 | 3, 7, 10 | 3eqtr4d 2123 | . . . . . 6 |
12 | simplr 496 | . . . . . . 7 | |
13 | 12 | fveq1d 5200 | . . . . . 6 |
14 | 12 | fveq1d 5200 | . . . . . 6 |
15 | 11, 13, 14 | 3eqtr3d 2121 | . . . . 5 |
16 | fvresi 5377 | . . . . . 6 | |
17 | 5, 16 | syl 14 | . . . . 5 |
18 | fvresi 5377 | . . . . . 6 | |
19 | 8, 18 | syl 14 | . . . . 5 |
20 | 15, 17, 19 | 3eqtr3d 2121 | . . . 4 |
21 | 20 | expr 367 | . . 3 |
22 | 21 | ralrimivva 2443 | . 2 |
23 | dff13 5428 | . 2 | |
24 | 1, 22, 23 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wral 2348 cid 4043 cres 4365 ccom 4367 wf 4918 wf1 4919 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fv 4930 |
This theorem is referenced by: fcof1o 5449 |
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