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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 5441 | . . . . . . . . 9 | |
3 | 2 | adantrr 462 | . . . . . . . 8 |
4 | 3 | 3adant3 958 | . . . . . . 7 |
5 | f1ocnvdm 5441 | . . . . . . . . . 10 | |
6 | 5 | adantrl 461 | . . . . . . . . 9 |
7 | 6 | 3adant3 958 | . . . . . . . 8 |
8 | f1ocnvfv2 5438 | . . . . . . . . . . 11 | |
9 | 8 | eqcomd 2086 | . . . . . . . . . 10 |
10 | f1ocnvfv2 5438 | . . . . . . . . . . 11 | |
11 | 10 | eqcomd 2086 | . . . . . . . . . 10 |
12 | 9, 11 | anim12dan 564 | . . . . . . . . 9 |
13 | 12 | 3adant3 958 | . . . . . . . 8 |
14 | simp3 940 | . . . . . . . 8 | |
15 | fveq2 5198 | . . . . . . . . . . . 12 | |
16 | 15 | eqeq2d 2092 | . . . . . . . . . . 11 |
17 | 16 | anbi2d 451 | . . . . . . . . . 10 |
18 | breq2 3789 | . . . . . . . . . 10 | |
19 | 17, 18 | anbi12d 456 | . . . . . . . . 9 |
20 | 19 | rspcev 2701 | . . . . . . . 8 |
21 | 7, 13, 14, 20 | syl12anc 1167 | . . . . . . 7 |
22 | fveq2 5198 | . . . . . . . . . . . 12 | |
23 | 22 | eqeq2d 2092 | . . . . . . . . . . 11 |
24 | 23 | anbi1d 452 | . . . . . . . . . 10 |
25 | breq1 3788 | . . . . . . . . . 10 | |
26 | 24, 25 | anbi12d 456 | . . . . . . . . 9 |
27 | 26 | rexbidv 2369 | . . . . . . . 8 |
28 | 27 | rspcev 2701 | . . . . . . 7 |
29 | 4, 21, 28 | syl2anc 403 | . . . . . 6 |
30 | 29 | 3expib 1141 | . . . . 5 |
31 | simp3ll 1009 | . . . . . . . . 9 | |
32 | simp1 938 | . . . . . . . . . 10 | |
33 | simp2l 964 | . . . . . . . . . 10 | |
34 | f1of 5146 | . . . . . . . . . . 11 | |
35 | 34 | ffvelrnda 5323 | . . . . . . . . . 10 |
36 | 32, 33, 35 | syl2anc 403 | . . . . . . . . 9 |
37 | 31, 36 | eqeltrd 2155 | . . . . . . . 8 |
38 | simp3lr 1010 | . . . . . . . . 9 | |
39 | simp2r 965 | . . . . . . . . . 10 | |
40 | 34 | ffvelrnda 5323 | . . . . . . . . . 10 |
41 | 32, 39, 40 | syl2anc 403 | . . . . . . . . 9 |
42 | 38, 41 | eqeltrd 2155 | . . . . . . . 8 |
43 | simp3r 967 | . . . . . . . . 9 | |
44 | 31 | eqcomd 2086 | . . . . . . . . . 10 |
45 | f1ocnvfv 5439 | . . . . . . . . . . 11 | |
46 | 32, 33, 45 | syl2anc 403 | . . . . . . . . . 10 |
47 | 44, 46 | mpd 13 | . . . . . . . . 9 |
48 | 38 | eqcomd 2086 | . . . . . . . . . 10 |
49 | f1ocnvfv 5439 | . . . . . . . . . . 11 | |
50 | 32, 39, 49 | syl2anc 403 | . . . . . . . . . 10 |
51 | 48, 50 | mpd 13 | . . . . . . . . 9 |
52 | 43, 47, 51 | 3brtr4d 3815 | . . . . . . . 8 |
53 | 37, 42, 52 | jca31 302 | . . . . . . 7 |
54 | 53 | 3exp 1137 | . . . . . 6 |
55 | 54 | rexlimdvv 2483 | . . . . 5 |
56 | 30, 55 | impbid 127 | . . . 4 |
57 | 56 | opabbidv 3844 | . . 3 |
58 | 1, 57 | syl5eq 2125 | . 2 |
59 | f1oiso 5485 | . 2 | |
60 | 58, 59 | mpdan 412 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785 copab 3838 ccnv 4362 wf1o 4921 cfv 4922 wiso 4923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-isom 4931 |
This theorem is referenced by: (None) |
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