Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 521 | . . . . . . . 8 | |
3 | 2 | fveq2d 5425 | . . . . . . 7 |
4 | simpll 518 | . . . . . . . 8 | |
5 | simprll 526 | . . . . . . . 8 | |
6 | fvco3 5492 | . . . . . . . 8 | |
7 | 4, 5, 6 | syl2anc 408 | . . . . . . 7 |
8 | simprlr 527 | . . . . . . . 8 | |
9 | fvco3 5492 | . . . . . . . 8 | |
10 | 4, 8, 9 | syl2anc 408 | . . . . . . 7 |
11 | 3, 7, 10 | 3eqtr4d 2182 | . . . . . 6 |
12 | simplr 519 | . . . . . . 7 | |
13 | 12 | fveq1d 5423 | . . . . . 6 |
14 | 12 | fveq1d 5423 | . . . . . 6 |
15 | 11, 13, 14 | 3eqtr3d 2180 | . . . . 5 |
16 | fvresi 5613 | . . . . . 6 | |
17 | 5, 16 | syl 14 | . . . . 5 |
18 | fvresi 5613 | . . . . . 6 | |
19 | 8, 18 | syl 14 | . . . . 5 |
20 | 15, 17, 19 | 3eqtr3d 2180 | . . . 4 |
21 | 20 | expr 372 | . . 3 |
22 | 21 | ralrimivva 2514 | . 2 |
23 | dff13 5669 | . 2 | |
24 | 1, 22, 23 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cid 4210 cres 4541 ccom 4543 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-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-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: fcof1o 5690 |
Copyright terms: Public domain | W3C validator |