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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 5854 |
. . . . . 6
| |
| 2 | f1of1 5503 |
. . . . . 6
| |
| 3 | 1, 2 | syl 14 |
. . . . 5
|
| 4 | 3 | adantr 276 |
. . . 4
|
| 5 | isoini2.1 |
. . . . 5
| |
| 6 | inss1 3383 |
. . . . 5
| |
| 7 | 5, 6 | eqsstri 3215 |
. . . 4
|
| 8 | f1ores 5519 |
. . . 4
| |
| 9 | 4, 7, 8 | sylancl 413 |
. . 3
|
| 10 | isoini 5865 |
. . . . 5
| |
| 11 | 5 | imaeq2i 5007 |
. . . . 5
|
| 12 | isoini2.2 |
. . . . 5
| |
| 13 | 10, 11, 12 | 3eqtr4g 2254 |
. . . 4
|
| 14 | f1oeq3 5494 |
. . . 4
| |
| 15 | 13, 14 | syl 14 |
. . 3
|
| 16 | 9, 15 | mpbid 147 |
. 2
|
| 17 | df-isom 5267 |
. . . . . . 7
| |
| 18 | 17 | simprbi 275 |
. . . . . 6
|
| 19 | 18 | adantr 276 |
. . . . 5
|
| 20 | ssralv 3247 |
. . . . . 6
| |
| 21 | 20 | ralimdv 2565 |
. . . . 5
|
| 22 | 7, 19, 21 | mpsyl 65 |
. . . 4
|
| 23 | ssralv 3247 |
. . . 4
| |
| 24 | 7, 22, 23 | mpsyl 65 |
. . 3
|
| 25 | fvres 5582 |
. . . . . . 7
| |
| 26 | fvres 5582 |
. . . . . . 7
| |
| 27 | 25, 26 | breqan12d 4049 |
. . . . . 6
|
| 28 | 27 | bibi2d 232 |
. . . . 5
|
| 29 | 28 | ralbidva 2493 |
. . . 4
|
| 30 | 29 | ralbiia 2511 |
. . 3
|
| 31 | 24, 30 | sylibr 134 |
. 2
|
| 32 | df-isom 5267 |
. 2
| |
| 33 | 16, 31, 32 | sylanbrc 417 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-sbc 2990 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 |
| This theorem is referenced by: (None) |
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