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Mirrors > Home > ILE Home > Th. List > isocnv2 | Unicode version |
Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Ref | Expression |
---|---|
isocnv2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isof1o 5467 | . . 3 | |
2 | f1ofn 5147 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | isof1o 5467 | . . 3 | |
5 | 4, 2 | syl 14 | . 2 |
6 | vex 2604 | . . . . . . . . . 10 | |
7 | vex 2604 | . . . . . . . . . 10 | |
8 | 6, 7 | brcnv 4536 | . . . . . . . . 9 |
9 | 8 | a1i 9 | . . . . . . . 8 |
10 | funfvex 5212 | . . . . . . . . . . 11 | |
11 | 10 | funfni 5019 | . . . . . . . . . 10 |
12 | 11 | adantr 270 | . . . . . . . . 9 |
13 | funfvex 5212 | . . . . . . . . . . 11 | |
14 | 13 | funfni 5019 | . . . . . . . . . 10 |
15 | 14 | adantlr 460 | . . . . . . . . 9 |
16 | brcnvg 4534 | . . . . . . . . 9 | |
17 | 12, 15, 16 | syl2anc 403 | . . . . . . . 8 |
18 | 9, 17 | bibi12d 233 | . . . . . . 7 |
19 | 18 | ralbidva 2364 | . . . . . 6 |
20 | 19 | ralbidva 2364 | . . . . 5 |
21 | ralcom 2517 | . . . . 5 | |
22 | 20, 21 | syl6rbbr 197 | . . . 4 |
23 | 22 | anbi2d 451 | . . 3 |
24 | df-isom 4931 | . . 3 | |
25 | df-isom 4931 | . . 3 | |
26 | 23, 24, 25 | 3bitr4g 221 | . 2 |
27 | 3, 5, 26 | pm5.21nii 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wcel 1433 wral 2348 cvv 2601 class class class wbr 3785 ccnv 4362 wfn 4917 wf1o 4921 cfv 4922 wiso 4923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-f1o 4929 df-fv 4930 df-isom 4931 |
This theorem is referenced by: infisoti 6445 |
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