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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 5366 | . . . . . . 7 | |
2 | f1ores 5382 | . . . . . . . 8 | |
3 | 2 | expcom 115 | . . . . . . 7 |
4 | 1, 3 | syl5 32 | . . . . . 6 |
5 | ssralv 3161 | . . . . . . 7 | |
6 | ssralv 3161 | . . . . . . . . . 10 | |
7 | 6 | adantr 274 | . . . . . . . . 9 |
8 | fvres 5445 | . . . . . . . . . . . . . 14 | |
9 | fvres 5445 | . . . . . . . . . . . . . 14 | |
10 | 8, 9 | breqan12d 3945 | . . . . . . . . . . . . 13 |
11 | 10 | adantll 467 | . . . . . . . . . . . 12 |
12 | 11 | bibi2d 231 | . . . . . . . . . . 11 |
13 | 12 | biimprd 157 | . . . . . . . . . 10 |
14 | 13 | ralimdva 2499 | . . . . . . . . 9 |
15 | 7, 14 | syld 45 | . . . . . . . 8 |
16 | 15 | ralimdva 2499 | . . . . . . 7 |
17 | 5, 16 | syld 45 | . . . . . 6 |
18 | 4, 17 | anim12d 333 | . . . . 5 |
19 | df-isom 5132 | . . . . 5 | |
20 | df-isom 5132 | . . . . 5 | |
21 | 18, 19, 20 | 3imtr4g 204 | . . . 4 |
22 | 21 | impcom 124 | . . 3 |
23 | isoeq5 5706 | . . 3 | |
24 | 22, 23 | syl5ibrcom 156 | . 2 |
25 | 24 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 wss 3071 class class class wbr 3929 cres 4541 cima 4542 wf1 5120 wf1o 5122 cfv 5123 wiso 5124 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 |
This theorem is referenced by: (None) |
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