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Mirrors > Home > MPE Home > Th. List > isoini2 | Structured version Visualization version 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 6573 | . . . . . 6 | |
2 | f1of1 6136 | . . . . . 6 | |
3 | 1, 2 | syl 17 | . . . . 5 |
4 | 3 | adantr 481 | . . . 4 |
5 | isoini2.1 | . . . . 5 | |
6 | inss1 3833 | . . . . 5 | |
7 | 5, 6 | eqsstri 3635 | . . . 4 |
8 | f1ores 6151 | . . . 4 | |
9 | 4, 7, 8 | sylancl 694 | . . 3 |
10 | isoini 6588 | . . . . 5 | |
11 | 5 | imaeq2i 5464 | . . . . 5 |
12 | isoini2.2 | . . . . 5 | |
13 | 10, 11, 12 | 3eqtr4g 2681 | . . . 4 |
14 | f1oeq3 6129 | . . . 4 | |
15 | 13, 14 | syl 17 | . . 3 |
16 | 9, 15 | mpbid 222 | . 2 |
17 | df-isom 5897 | . . . . . . 7 | |
18 | 17 | simprbi 480 | . . . . . 6 |
19 | 18 | adantr 481 | . . . . 5 |
20 | ssralv 3666 | . . . . . 6 | |
21 | 20 | ralimdv 2963 | . . . . 5 |
22 | 7, 19, 21 | mpsyl 68 | . . . 4 |
23 | ssralv 3666 | . . . 4 | |
24 | 7, 22, 23 | mpsyl 68 | . . 3 |
25 | fvres 6207 | . . . . . . 7 | |
26 | fvres 6207 | . . . . . . 7 | |
27 | 25, 26 | breqan12d 4669 | . . . . . 6 |
28 | 27 | bibi2d 332 | . . . . 5 |
29 | 28 | ralbidva 2985 | . . . 4 |
30 | 29 | ralbiia 2979 | . . 3 |
31 | 24, 30 | sylibr 224 | . 2 |
32 | df-isom 5897 | . 2 | |
33 | 16, 31, 32 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cin 3573 wss 3574 csn 4177 class class class wbr 4653 ccnv 5113 cres 5116 cima 5117 wf1 5885 wf1o 5887 cfv 5888 wiso 5889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 |
This theorem is referenced by: fz1isolem 13245 |
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