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Mirrors > Home > ILE Home > Th. List > isoselem | Unicode version |
Description: Lemma for isose 5460. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Ref | Expression |
---|---|
isofrlem.1 | |
isofrlem.2 |
Ref | Expression |
---|---|
isoselem | Se Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfse2 4698 | . . . . . . . . 9 Se | |
2 | 1 | biimpi 113 | . . . . . . . 8 Se |
3 | 2 | r19.21bi 2407 | . . . . . . 7 Se |
4 | 3 | expcom 109 | . . . . . 6 Se |
5 | 4 | adantl 262 | . . . . 5 Se |
6 | imaeq2 4664 | . . . . . . . . . . 11 | |
7 | 6 | eleq1d 2106 | . . . . . . . . . 10 |
8 | 7 | imbi2d 219 | . . . . . . . . 9 |
9 | isofrlem.2 | . . . . . . . . 9 | |
10 | 8, 9 | vtoclg 2613 | . . . . . . . 8 |
11 | 10 | com12 27 | . . . . . . 7 |
12 | 11 | adantr 261 | . . . . . 6 |
13 | isofrlem.1 | . . . . . . . 8 | |
14 | isoini 5457 | . . . . . . . 8 | |
15 | 13, 14 | sylan 267 | . . . . . . 7 |
16 | 15 | eleq1d 2106 | . . . . . 6 |
17 | 12, 16 | sylibd 138 | . . . . 5 |
18 | 5, 17 | syld 40 | . . . 4 Se |
19 | 18 | ralrimdva 2399 | . . 3 Se |
20 | isof1o 5447 | . . . . 5 | |
21 | f1ofn 5127 | . . . . 5 | |
22 | sneq 3386 | . . . . . . . . 9 | |
23 | 22 | imaeq2d 4668 | . . . . . . . 8 |
24 | 23 | ineq2d 3138 | . . . . . . 7 |
25 | 24 | eleq1d 2106 | . . . . . 6 |
26 | 25 | ralrn 5305 | . . . . 5 |
27 | 13, 20, 21, 26 | 4syl 18 | . . . 4 |
28 | f1ofo 5133 | . . . . . 6 | |
29 | forn 5109 | . . . . . 6 | |
30 | 13, 20, 28, 29 | 4syl 18 | . . . . 5 |
31 | 30 | raleqdv 2511 | . . . 4 |
32 | 27, 31 | bitr3d 179 | . . 3 |
33 | 19, 32 | sylibd 138 | . 2 Se |
34 | dfse2 4698 | . 2 Se | |
35 | 33, 34 | syl6ibr 151 | 1 Se Se |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 cin 2916 csn 3375 Se wse 4066 ccnv 4344 crn 4346 cima 4348 wfn 4897 wfo 4900 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-rab 2315 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-se 4070 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: isose 5460 |
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