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Mirrors > Home > ILE Home > Th. List > isores3 | Unicode version |
Description: Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Ref | Expression |
---|---|
isores3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1of1 5145 | . . . . . . 7 | |
2 | f1ores 5161 | . . . . . . . 8 | |
3 | 2 | expcom 114 | . . . . . . 7 |
4 | 1, 3 | syl5 32 | . . . . . 6 |
5 | ssralv 3058 | . . . . . . 7 | |
6 | ssralv 3058 | . . . . . . . . . 10 | |
7 | 6 | adantr 270 | . . . . . . . . 9 |
8 | fvres 5219 | . . . . . . . . . . . . . 14 | |
9 | fvres 5219 | . . . . . . . . . . . . . 14 | |
10 | 8, 9 | breqan12d 3800 | . . . . . . . . . . . . 13 |
11 | 10 | adantll 459 | . . . . . . . . . . . 12 |
12 | 11 | bibi2d 230 | . . . . . . . . . . 11 |
13 | 12 | biimprd 156 | . . . . . . . . . 10 |
14 | 13 | ralimdva 2429 | . . . . . . . . 9 |
15 | 7, 14 | syld 44 | . . . . . . . 8 |
16 | 15 | ralimdva 2429 | . . . . . . 7 |
17 | 5, 16 | syld 44 | . . . . . 6 |
18 | 4, 17 | anim12d 328 | . . . . 5 |
19 | df-isom 4931 | . . . . 5 | |
20 | df-isom 4931 | . . . . 5 | |
21 | 18, 19, 20 | 3imtr4g 203 | . . . 4 |
22 | 21 | impcom 123 | . . 3 |
23 | isoeq5 5465 | . . 3 | |
24 | 22, 23 | syl5ibrcom 155 | . 2 |
25 | 24 | 3impia 1135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wral 2348 wss 2973 class class class wbr 3785 cres 4365 cima 4366 wf1 4919 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-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 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-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-isom 4931 |
This theorem is referenced by: (None) |
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