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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 5334 | . . . . . . 7 | |
2 | f1ores 5350 | . . . . . . . 8 | |
3 | 2 | expcom 115 | . . . . . . 7 |
4 | 1, 3 | syl5 32 | . . . . . 6 |
5 | ssralv 3131 | . . . . . . 7 | |
6 | ssralv 3131 | . . . . . . . . . 10 | |
7 | 6 | adantr 274 | . . . . . . . . 9 |
8 | fvres 5413 | . . . . . . . . . . . . . 14 | |
9 | fvres 5413 | . . . . . . . . . . . . . 14 | |
10 | 8, 9 | breqan12d 3915 | . . . . . . . . . . . . 13 |
11 | 10 | adantll 467 | . . . . . . . . . . . 12 |
12 | 11 | bibi2d 231 | . . . . . . . . . . 11 |
13 | 12 | biimprd 157 | . . . . . . . . . 10 |
14 | 13 | ralimdva 2476 | . . . . . . . . 9 |
15 | 7, 14 | syld 45 | . . . . . . . 8 |
16 | 15 | ralimdva 2476 | . . . . . . 7 |
17 | 5, 16 | syld 45 | . . . . . 6 |
18 | 4, 17 | anim12d 333 | . . . . 5 |
19 | df-isom 5102 | . . . . 5 | |
20 | df-isom 5102 | . . . . 5 | |
21 | 18, 19, 20 | 3imtr4g 204 | . . . 4 |
22 | 21 | impcom 124 | . . 3 |
23 | isoeq5 5674 | . . 3 | |
24 | 22, 23 | syl5ibrcom 156 | . 2 |
25 | 24 | 3impia 1163 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wral 2393 wss 3041 class class class wbr 3899 cres 4511 cima 4512 wf1 5090 wf1o 5092 cfv 5093 wiso 5094 |
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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 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-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-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 |
This theorem is referenced by: (None) |
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