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Theorem fcofo 6002
Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcofo

Proof of Theorem fcofo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 962 . 2
2 ffvelrn 5857 . . . . 5
323ad2antl2 1125 . . . 4
4 simpl3 967 . . . . . 6
54fveq1d 5710 . . . . 5
6 fvco3 5784 . . . . . 6
763ad2antl2 1125 . . . . 5
8 fvresi 5919 . . . . . 6
98adantl 454 . . . . 5
105, 7, 93eqtr3rd 2530 . . . 4
11 fveq2 5708 . . . . . 6
1211eqeq2d 2500 . . . . 5
1312rspcev 3113 . . . 4
143, 10, 13syl2anc 644 . . 3
1514ralrimiva 2843 . 2
16 dffo3 5874 . 2
171, 15, 16sylanbrc 647 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  /\w3a 939  =wceq 1670  e.wcel 1732  A.wral 2759  E.wrex 2760   cid 4652  |`cres 4864  o.ccom 4866  -->wf 5434  -onto->wfo 5436  `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-mpt 4378  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-fo 5444  df-fv 5446
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