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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 5125 | . . . . . . 7 | |
2 | f1ores 5141 | . . . . . . . 8 | |
3 | 2 | expcom 109 | . . . . . . 7 |
4 | 1, 3 | syl5 28 | . . . . . 6 |
5 | ssralv 3004 | . . . . . . 7 | |
6 | ssralv 3004 | . . . . . . . . . 10 | |
7 | 6 | adantr 261 | . . . . . . . . 9 |
8 | fvres 5198 | . . . . . . . . . . . . . 14 | |
9 | fvres 5198 | . . . . . . . . . . . . . 14 | |
10 | 8, 9 | breqan12d 3779 | . . . . . . . . . . . . 13 |
11 | 10 | adantll 445 | . . . . . . . . . . . 12 |
12 | 11 | bibi2d 221 | . . . . . . . . . . 11 |
13 | 12 | biimprd 147 | . . . . . . . . . 10 |
14 | 13 | ralimdva 2387 | . . . . . . . . 9 |
15 | 7, 14 | syld 40 | . . . . . . . 8 |
16 | 15 | ralimdva 2387 | . . . . . . 7 |
17 | 5, 16 | syld 40 | . . . . . 6 |
18 | 4, 17 | anim12d 318 | . . . . 5 |
19 | df-isom 4911 | . . . . 5 | |
20 | df-isom 4911 | . . . . 5 | |
21 | 18, 19, 20 | 3imtr4g 194 | . . . 4 |
22 | 21 | impcom 116 | . . 3 |
23 | isoeq5 5445 | . . 3 | |
24 | 22, 23 | syl5ibrcom 146 | . 2 |
25 | 24 | 3impia 1101 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 wral 2306 wss 2917 class class class wbr 3764 cres 4347 cima 4348 wf1 4899 wf1o 4901 cfv 4902 wiso 4903 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-isom 4911 |
This theorem is referenced by: (None) |
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