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Mirrors > Home > MPE Home > Th. List > fcofo | Unicode version |
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.) |
Ref | Expression |
---|---|
fcofo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 996 | . 2 | |
2 | ffvelrn 6029 | . . . . 5 | |
3 | 2 | 3ad2antl2 1159 | . . . 4 |
4 | simpl3 1001 | . . . . . 6 | |
5 | 4 | fveq1d 5873 | . . . . 5 |
6 | fvco3 5950 | . . . . . 6 | |
7 | 6 | 3ad2antl2 1159 | . . . . 5 |
8 | fvresi 6097 | . . . . . 6 | |
9 | 8 | adantl 466 | . . . . 5 |
10 | 5, 7, 9 | 3eqtr3rd 2507 | . . . 4 |
11 | fveq2 5871 | . . . . . 6 | |
12 | 11 | eqeq2d 2471 | . . . . 5 |
13 | 12 | rspcev 3210 | . . . 4 |
14 | 3, 10, 13 | syl2anc 661 | . . 3 |
15 | 14 | ralrimiva 2871 | . 2 |
16 | dffo3 6046 | . 2 | |
17 | 1, 15, 16 | sylanbrc 664 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
/\ w3a 973 = wceq 1395 e. wcel 1818
A. wral 2807 E. wrex 2808 cid 4795
|` cres 5006 o. ccom 5008 --> wf 5589
-onto-> wfo 5591
` 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-mpt 4512 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-fo 5599 df-fv 5601 |
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