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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 108 | . 2 | |
2 | simprr 506 | . . . . . . . 8 | |
3 | 2 | fveq2d 5393 | . . . . . . 7 |
4 | simpll 503 | . . . . . . . 8 | |
5 | simprll 511 | . . . . . . . 8 | |
6 | fvco3 5460 | . . . . . . . 8 | |
7 | 4, 5, 6 | syl2anc 408 | . . . . . . 7 |
8 | simprlr 512 | . . . . . . . 8 | |
9 | fvco3 5460 | . . . . . . . 8 | |
10 | 4, 8, 9 | syl2anc 408 | . . . . . . 7 |
11 | 3, 7, 10 | 3eqtr4d 2160 | . . . . . 6 |
12 | simplr 504 | . . . . . . 7 | |
13 | 12 | fveq1d 5391 | . . . . . 6 |
14 | 12 | fveq1d 5391 | . . . . . 6 |
15 | 11, 13, 14 | 3eqtr3d 2158 | . . . . 5 |
16 | fvresi 5581 | . . . . . 6 | |
17 | 5, 16 | syl 14 | . . . . 5 |
18 | fvresi 5581 | . . . . . 6 | |
19 | 8, 18 | syl 14 | . . . . 5 |
20 | 15, 17, 19 | 3eqtr3d 2158 | . . . 4 |
21 | 20 | expr 372 | . . 3 |
22 | 21 | ralrimivva 2491 | . 2 |
23 | dff13 5637 | . 2 | |
24 | 1, 22, 23 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wral 2393 cid 4180 cres 4511 ccom 4513 wf 5089 wf1 5090 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fv 5101 |
This theorem is referenced by: fcof1o 5658 |
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