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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 992 | . 2 | |
2 | ffvelrn 5626 | . . . . 5 | |
3 | 2 | 3ad2antl2 1155 | . . . 4 |
4 | simpl3 997 | . . . . . 6 | |
5 | 4 | fveq1d 5496 | . . . . 5 |
6 | fvco3 5565 | . . . . . 6 | |
7 | 6 | 3ad2antl2 1155 | . . . . 5 |
8 | fvresi 5686 | . . . . . 6 | |
9 | 8 | adantl 275 | . . . . 5 |
10 | 5, 7, 9 | 3eqtr3rd 2212 | . . . 4 |
11 | fveq2 5494 | . . . . . 6 | |
12 | 11 | eqeq2d 2182 | . . . . 5 |
13 | 12 | rspcev 2834 | . . . 4 |
14 | 3, 10, 13 | syl2anc 409 | . . 3 |
15 | 14 | ralrimiva 2543 | . 2 |
16 | dffo3 5640 | . 2 | |
17 | 1, 15, 16 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 wral 2448 wrex 2449 cid 4271 cres 4611 ccom 4613 wf 5192 wfo 5194 cfv 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fo 5202 df-fv 5204 |
This theorem is referenced by: fcof1o 5765 |
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