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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 5676 | . . . . . 6 | |
2 | f1of1 5334 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | isoini2.1 | . . . . 5 | |
6 | inss1 3266 | . . . . 5 | |
7 | 5, 6 | eqsstri 3099 | . . . 4 |
8 | f1ores 5350 | . . . 4 | |
9 | 4, 7, 8 | sylancl 409 | . . 3 |
10 | isoini 5687 | . . . . 5 | |
11 | 5 | imaeq2i 4849 | . . . . 5 |
12 | isoini2.2 | . . . . 5 | |
13 | 10, 11, 12 | 3eqtr4g 2175 | . . . 4 |
14 | f1oeq3 5328 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 9, 15 | mpbid 146 | . 2 |
17 | df-isom 5102 | . . . . . . 7 | |
18 | 17 | simprbi 273 | . . . . . 6 |
19 | 18 | adantr 274 | . . . . 5 |
20 | ssralv 3131 | . . . . . 6 | |
21 | 20 | ralimdv 2477 | . . . . 5 |
22 | 7, 19, 21 | mpsyl 65 | . . . 4 |
23 | ssralv 3131 | . . . 4 | |
24 | 7, 22, 23 | mpsyl 65 | . . 3 |
25 | fvres 5413 | . . . . . . 7 | |
26 | fvres 5413 | . . . . . . 7 | |
27 | 25, 26 | breqan12d 3915 | . . . . . 6 |
28 | 27 | bibi2d 231 | . . . . 5 |
29 | 28 | ralbidva 2410 | . . . 4 |
30 | 29 | ralbiia 2426 | . . 3 |
31 | 24, 30 | sylibr 133 | . 2 |
32 | df-isom 5102 | . 2 | |
33 | 16, 31, 32 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 cin 3040 wss 3041 csn 3497 class class class wbr 3899 ccnv 4508 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-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 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-mpt 3961 df-id 4185 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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