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Theorem fcof1 6001
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 445 . 2
2 simprr 735 . . . . . . . 8
32fveq2d 5712 . . . . . . 7
4 simpll 732 . . . . . . . 8
5 simprll 740 . . . . . . . 8
6 fvco3 5784 . . . . . . . 8
74, 5, 6syl2anc 644 . . . . . . 7
8 simprlr 741 . . . . . . . 8
9 fvco3 5784 . . . . . . . 8
104, 8, 9syl2anc 644 . . . . . . 7
113, 7, 103eqtr4d 2531 . . . . . 6
12 simplr 733 . . . . . . 7
1312fveq1d 5710 . . . . . 6
1412fveq1d 5710 . . . . . 6
1511, 13, 143eqtr3d 2529 . . . . 5
16 fvresi 5919 . . . . . 6
175, 16syl 16 . . . . 5
18 fvresi 5919 . . . . . 6
198, 18syl 16 . . . . 5
2015, 17, 193eqtr3d 2529 . . . 4
2120expr 600 . . 3
2221ralrimivva 2852 . 2
23 dff13 5985 . 2
241, 22, 23sylanbrc 647 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  =wceq 1670  e.wcel 1732  A.wral 2759   cid 4652  |`cres 4864  o.ccom 4866  -->wf 5434  -1-1->wf1 5435  `cfv 5438
This theorem is referenced by:  fcof1o  6007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1570  ax-4 1581  ax-5 1644  ax-6 1685  ax-7 1705  ax-8 1734  ax-9 1736  ax-10 1751  ax-11 1756  ax-12 1768  ax-13 1955  ax-ext 2470  ax-sep 4439  ax-nul 4447  ax-pow 4493  ax-pr 4554
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1338  df-ex 1566  df-nf 1569  df-sb 1677  df-eu 2317  df-mo 2318  df-clab 2476  df-cleq 2482  df-clel 2485  df-nfc 2614  df-ne 2654  df-ral 2764  df-rex 2765  df-rab 2768  df-v 3017  df-sbc 3225  df-dif 3368  df-un 3370  df-in 3372  df-ss 3379  df-nul 3674  df-if 3826  df-sn 3915  df-pr 3916  df-op 3918  df-uni 4118  df-br 4319  df-opab 4377  df-id 4657  df-xp 4868  df-rel 4869  df-cnv 4870  df-co 4871  df-dm 4872  df-rn 4873  df-res 4874  df-ima 4875  df-iota 5401  df-fun 5440  df-fn 5441  df-f 5442  df-f1 5443  df-fv 5446
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