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Theorem fcof1 6011
 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 444 . 2
2 simprr 734 . . . . . . . 8
32fveq2d 5723 . . . . . . 7
4 simpll 731 . . . . . . . 8
5 simprll 739 . . . . . . . 8
6 fvco3 5791 . . . . . . . 8
74, 5, 6syl2anc 643 . . . . . . 7
8 simprlr 740 . . . . . . . 8
9 fvco3 5791 . . . . . . . 8
104, 8, 9syl2anc 643 . . . . . . 7
113, 7, 103eqtr4d 2477 . . . . . 6
12 simplr 732 . . . . . . 7
1312fveq1d 5721 . . . . . 6
1412fveq1d 5721 . . . . . 6
1511, 13, 143eqtr3d 2475 . . . . 5
16 fvresi 5915 . . . . . 6
175, 16syl 16 . . . . 5
18 fvresi 5915 . . . . . 6
198, 18syl 16 . . . . 5
2015, 17, 193eqtr3d 2475 . . . 4
2120expr 599 . . 3
2221ralrimivva 2790 . 2
23 dff13 5995 . 2
241, 22, 23sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697   cid 4485   cres 4871   ccom 4873  wf 5441  wf1 5442  cfv 5445 This theorem is referenced by:  fcof1o  6017 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fv 5453
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