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Mirrors > Home > ILE Home > Th. List > f1oiso2 | Unicode version |
Description: Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.) |
Ref | Expression |
---|---|
f1oiso2.1 |
Ref | Expression |
---|---|
f1oiso2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oiso2.1 | . . 3 | |
2 | f1ocnvdm 5421 | . . . . . . . . 9 | |
3 | 2 | adantrr 448 | . . . . . . . 8 |
4 | 3 | 3adant3 924 | . . . . . . 7 |
5 | f1ocnvdm 5421 | . . . . . . . . . 10 | |
6 | 5 | adantrl 447 | . . . . . . . . 9 |
7 | 6 | 3adant3 924 | . . . . . . . 8 |
8 | f1ocnvfv2 5418 | . . . . . . . . . . 11 | |
9 | 8 | eqcomd 2045 | . . . . . . . . . 10 |
10 | f1ocnvfv2 5418 | . . . . . . . . . . 11 | |
11 | 10 | eqcomd 2045 | . . . . . . . . . 10 |
12 | 9, 11 | anim12dan 532 | . . . . . . . . 9 |
13 | 12 | 3adant3 924 | . . . . . . . 8 |
14 | simp3 906 | . . . . . . . 8 | |
15 | fveq2 5178 | . . . . . . . . . . . 12 | |
16 | 15 | eqeq2d 2051 | . . . . . . . . . . 11 |
17 | 16 | anbi2d 437 | . . . . . . . . . 10 |
18 | breq2 3768 | . . . . . . . . . 10 | |
19 | 17, 18 | anbi12d 442 | . . . . . . . . 9 |
20 | 19 | rspcev 2656 | . . . . . . . 8 |
21 | 7, 13, 14, 20 | syl12anc 1133 | . . . . . . 7 |
22 | fveq2 5178 | . . . . . . . . . . . 12 | |
23 | 22 | eqeq2d 2051 | . . . . . . . . . . 11 |
24 | 23 | anbi1d 438 | . . . . . . . . . 10 |
25 | breq1 3767 | . . . . . . . . . 10 | |
26 | 24, 25 | anbi12d 442 | . . . . . . . . 9 |
27 | 26 | rexbidv 2327 | . . . . . . . 8 |
28 | 27 | rspcev 2656 | . . . . . . 7 |
29 | 4, 21, 28 | syl2anc 391 | . . . . . 6 |
30 | 29 | 3expib 1107 | . . . . 5 |
31 | simp3ll 975 | . . . . . . . . 9 | |
32 | simp1 904 | . . . . . . . . . 10 | |
33 | simp2l 930 | . . . . . . . . . 10 | |
34 | f1of 5126 | . . . . . . . . . . 11 | |
35 | 34 | ffvelrnda 5302 | . . . . . . . . . 10 |
36 | 32, 33, 35 | syl2anc 391 | . . . . . . . . 9 |
37 | 31, 36 | eqeltrd 2114 | . . . . . . . 8 |
38 | simp3lr 976 | . . . . . . . . 9 | |
39 | simp2r 931 | . . . . . . . . . 10 | |
40 | 34 | ffvelrnda 5302 | . . . . . . . . . 10 |
41 | 32, 39, 40 | syl2anc 391 | . . . . . . . . 9 |
42 | 38, 41 | eqeltrd 2114 | . . . . . . . 8 |
43 | simp3r 933 | . . . . . . . . 9 | |
44 | 31 | eqcomd 2045 | . . . . . . . . . 10 |
45 | f1ocnvfv 5419 | . . . . . . . . . . 11 | |
46 | 32, 33, 45 | syl2anc 391 | . . . . . . . . . 10 |
47 | 44, 46 | mpd 13 | . . . . . . . . 9 |
48 | 38 | eqcomd 2045 | . . . . . . . . . 10 |
49 | f1ocnvfv 5419 | . . . . . . . . . . 11 | |
50 | 32, 39, 49 | syl2anc 391 | . . . . . . . . . 10 |
51 | 48, 50 | mpd 13 | . . . . . . . . 9 |
52 | 43, 47, 51 | 3brtr4d 3794 | . . . . . . . 8 |
53 | 37, 42, 52 | jca31 292 | . . . . . . 7 |
54 | 53 | 3exp 1103 | . . . . . 6 |
55 | 54 | rexlimdvv 2439 | . . . . 5 |
56 | 30, 55 | impbid 120 | . . . 4 |
57 | 56 | opabbidv 3823 | . . 3 |
58 | 1, 57 | syl5eq 2084 | . 2 |
59 | f1oiso 5465 | . 2 | |
60 | 58, 59 | mpdan 398 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wrex 2307 class class class wbr 3764 copab 3817 ccnv 4344 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-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-id 4030 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: (None) |
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