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Mirrors > Home > ILE Home > Th. List > isoini2 | Unicode version |
Description: Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
Ref | Expression |
---|---|
isoini2.1 | |
isoini2.2 |
Ref | Expression |
---|---|
isoini2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isof1o 5447 | . . . . . 6 | |
2 | f1of1 5125 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantr 261 | . . . 4 |
5 | isoini2.1 | . . . . 5 | |
6 | inss1 3157 | . . . . 5 | |
7 | 5, 6 | eqsstri 2975 | . . . 4 |
8 | f1ores 5141 | . . . 4 | |
9 | 4, 7, 8 | sylancl 392 | . . 3 |
10 | isoini 5457 | . . . . 5 | |
11 | 5 | imaeq2i 4666 | . . . . 5 |
12 | isoini2.2 | . . . . 5 | |
13 | 10, 11, 12 | 3eqtr4g 2097 | . . . 4 |
14 | f1oeq3 5119 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 9, 15 | mpbid 135 | . 2 |
17 | df-isom 4911 | . . . . . . 7 | |
18 | 17 | simprbi 260 | . . . . . 6 |
19 | 18 | adantr 261 | . . . . 5 |
20 | ssralv 3004 | . . . . . 6 | |
21 | 20 | ralimdv 2388 | . . . . 5 |
22 | 7, 19, 21 | mpsyl 59 | . . . 4 |
23 | ssralv 3004 | . . . 4 | |
24 | 7, 22, 23 | mpsyl 59 | . . 3 |
25 | fvres 5198 | . . . . . . 7 | |
26 | fvres 5198 | . . . . . . 7 | |
27 | 25, 26 | breqan12d 3779 | . . . . . 6 |
28 | 27 | bibi2d 221 | . . . . 5 |
29 | 28 | ralbidva 2322 | . . . 4 |
30 | 29 | ralbiia 2338 | . . 3 |
31 | 24, 30 | sylibr 137 | . 2 |
32 | df-isom 4911 | . 2 | |
33 | 16, 31, 32 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cin 2916 wss 2917 csn 3375 class class class wbr 3764 ccnv 4344 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-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 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-mpt 3820 df-id 4030 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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