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Theorem fcof1 6190
 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.)
Assertion
Ref Expression
fcof1

Proof of Theorem fcof1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . 2
2 simprr 757 . . . . . . . 8
32fveq2d 5875 . . . . . . 7
4 simpll 753 . . . . . . . 8
5 simprll 763 . . . . . . . 8
6 fvco3 5950 . . . . . . . 8
74, 5, 6syl2anc 661 . . . . . . 7
8 simprlr 764 . . . . . . . 8
9 fvco3 5950 . . . . . . . 8
104, 8, 9syl2anc 661 . . . . . . 7
113, 7, 103eqtr4d 2508 . . . . . 6
12 simplr 755 . . . . . . 7
1312fveq1d 5873 . . . . . 6
1412fveq1d 5873 . . . . . 6
1511, 13, 143eqtr3d 2506 . . . . 5
16 fvresi 6097 . . . . . 6
175, 16syl 16 . . . . 5
18 fvresi 6097 . . . . . 6
198, 18syl 16 . . . . 5
2015, 17, 193eqtr3d 2506 . . . 4
2120expr 615 . . 3
2221ralrimivva 2878 . 2
23 dff13 6166 . 2
241, 22, 23sylanbrc 664 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807   cid 4795  |cres 5006  o.ccom 5008  -->wf 5589  -1-1->wf1 5590  cfv 5593 This theorem is referenced by:  fcof1od  6197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fv 5601
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