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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 5708 | . . . . . 6 | |
2 | f1of1 5366 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | isoini2.1 | . . . . 5 | |
6 | inss1 3296 | . . . . 5 | |
7 | 5, 6 | eqsstri 3129 | . . . 4 |
8 | f1ores 5382 | . . . 4 | |
9 | 4, 7, 8 | sylancl 409 | . . 3 |
10 | isoini 5719 | . . . . 5 | |
11 | 5 | imaeq2i 4879 | . . . . 5 |
12 | isoini2.2 | . . . . 5 | |
13 | 10, 11, 12 | 3eqtr4g 2197 | . . . 4 |
14 | f1oeq3 5358 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 9, 15 | mpbid 146 | . 2 |
17 | df-isom 5132 | . . . . . . 7 | |
18 | 17 | simprbi 273 | . . . . . 6 |
19 | 18 | adantr 274 | . . . . 5 |
20 | ssralv 3161 | . . . . . 6 | |
21 | 20 | ralimdv 2500 | . . . . 5 |
22 | 7, 19, 21 | mpsyl 65 | . . . 4 |
23 | ssralv 3161 | . . . 4 | |
24 | 7, 22, 23 | mpsyl 65 | . . 3 |
25 | fvres 5445 | . . . . . . 7 | |
26 | fvres 5445 | . . . . . . 7 | |
27 | 25, 26 | breqan12d 3945 | . . . . . 6 |
28 | 27 | bibi2d 231 | . . . . 5 |
29 | 28 | ralbidva 2433 | . . . 4 |
30 | 29 | ralbiia 2449 | . . 3 |
31 | 24, 30 | sylibr 133 | . 2 |
32 | df-isom 5132 | . 2 | |
33 | 16, 31, 32 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 cin 3070 wss 3071 csn 3527 class class class wbr 3929 ccnv 4538 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-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 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-mpt 3991 df-id 4215 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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