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Theorem fcofo 5966
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 5830 . . . . 5
323ad2antl2 1125 . . . 4
4 simpl3 967 . . . . . 6
54fveq1d 5687 . . . . 5
6 fvco3 5761 . . . . . 6
763ad2antl2 1125 . . . . 5
8 fvresi 5889 . . . . . 6
98adantl 454 . . . . 5
105, 7, 93eqtr3rd 2522 . . . 4
11 fveq2 5685 . . . . . 6
1211eqeq2d 2492 . . . . 5
1312rspcev 3102 . . . 4
143, 10, 13syl2anc 644 . . 3
1514ralrimiva 2835 . 2
16 dffo3 5846 . 2
171, 15, 16sylanbrc 647 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  /\wa 360  /\w3a 939  =wceq 1662  e.wcel 1724  A.wral 2751  E.wrex 2752   cid 4634  |`cres 4846  o.ccom 4848  -->wf 5413  -onto->wfo 5415  `cfv 5417
This theorem is referenced by:  fcof1o  5971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1562  ax-4 1573  ax-5 1636  ax-6 1677  ax-7 1697  ax-8 1726  ax-9 1728  ax-10 1743  ax-11 1748  ax-12 1760  ax-13 1947  ax-ext 2462  ax-sep 4423  ax-nul 4431  ax-pow 4477  ax-pr 4538
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1337  df-ex 1558  df-nf 1561  df-sb 1669  df-eu 2309  df-mo 2310  df-clab 2468  df-cleq 2474  df-clel 2477  df-nfc 2606  df-ne 2646  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3008  df-sbc 3213  df-dif 3356  df-un 3358  df-in 3360  df-ss 3367  df-nul 3661  df-if 3813  df-sn 3900  df-pr 3901  df-op 3903  df-uni 4102  df-br 4303  df-opab 4361  df-mpt 4362  df-id 4639  df-xp 4850  df-rel 4851  df-cnv 4852  df-co 4853  df-dm 4854  df-rn 4855  df-res 4856  df-ima 4857  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425
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