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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 102 | . 2 | |
2 | simprr 484 | . . . . . . . 8 | |
3 | 2 | fveq2d 5182 | . . . . . . 7 |
4 | simpll 481 | . . . . . . . 8 | |
5 | simprll 489 | . . . . . . . 8 | |
6 | fvco3 5244 | . . . . . . . 8 | |
7 | 4, 5, 6 | syl2anc 391 | . . . . . . 7 |
8 | simprlr 490 | . . . . . . . 8 | |
9 | fvco3 5244 | . . . . . . . 8 | |
10 | 4, 8, 9 | syl2anc 391 | . . . . . . 7 |
11 | 3, 7, 10 | 3eqtr4d 2082 | . . . . . 6 |
12 | simplr 482 | . . . . . . 7 | |
13 | 12 | fveq1d 5180 | . . . . . 6 |
14 | 12 | fveq1d 5180 | . . . . . 6 |
15 | 11, 13, 14 | 3eqtr3d 2080 | . . . . 5 |
16 | fvresi 5356 | . . . . . 6 | |
17 | 5, 16 | syl 14 | . . . . 5 |
18 | fvresi 5356 | . . . . . 6 | |
19 | 8, 18 | syl 14 | . . . . 5 |
20 | 15, 17, 19 | 3eqtr3d 2080 | . . . 4 |
21 | 20 | expr 357 | . . 3 |
22 | 21 | ralrimivva 2401 | . 2 |
23 | dff13 5407 | . 2 | |
24 | 1, 22, 23 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 cid 4025 cres 4347 ccom 4349 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-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-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: fcof1o 5429 |
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