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Mirrors > Home > ILE 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 966 | . 2 | |
2 | ffvelrn 5521 | . . . . 5 | |
3 | 2 | 3ad2antl2 1129 | . . . 4 |
4 | simpl3 971 | . . . . . 6 | |
5 | 4 | fveq1d 5391 | . . . . 5 |
6 | fvco3 5460 | . . . . . 6 | |
7 | 6 | 3ad2antl2 1129 | . . . . 5 |
8 | fvresi 5581 | . . . . . 6 | |
9 | 8 | adantl 275 | . . . . 5 |
10 | 5, 7, 9 | 3eqtr3rd 2159 | . . . 4 |
11 | fveq2 5389 | . . . . . 6 | |
12 | 11 | eqeq2d 2129 | . . . . 5 |
13 | 12 | rspcev 2763 | . . . 4 |
14 | 3, 10, 13 | syl2anc 408 | . . 3 |
15 | 14 | ralrimiva 2482 | . 2 |
16 | dffo3 5535 | . 2 | |
17 | 1, 15, 16 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wral 2393 wrex 2394 cid 4180 cres 4511 ccom 4513 wf 5089 wfo 5091 cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 |
This theorem is referenced by: fcof1o 5658 |
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