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Mirrors > Home > NFE 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 5481 | . . . . . . . . 9 | |
3 | 2 | adantrr 697 | . . . . . . . 8 |
4 | 3 | 3adant3 975 | . . . . . . 7 |
5 | f1ocnvdm 5481 | . . . . . . . . . 10 | |
6 | 5 | adantrl 696 | . . . . . . . . 9 |
7 | 6 | 3adant3 975 | . . . . . . . 8 |
8 | f1ocnvfv2 5477 | . . . . . . . . . . 11 | |
9 | 8 | eqcomd 2358 | . . . . . . . . . 10 |
10 | f1ocnvfv2 5477 | . . . . . . . . . . 11 | |
11 | 10 | eqcomd 2358 | . . . . . . . . . 10 |
12 | 9, 11 | anim12dan 810 | . . . . . . . . 9 |
13 | 12 | 3adant3 975 | . . . . . . . 8 |
14 | simp3 957 | . . . . . . . 8 | |
15 | fveq2 5328 | . . . . . . . . . . . 12 | |
16 | 15 | eqeq2d 2364 | . . . . . . . . . . 11 |
17 | 16 | anbi2d 684 | . . . . . . . . . 10 |
18 | breq2 4643 | . . . . . . . . . 10 | |
19 | 17, 18 | anbi12d 691 | . . . . . . . . 9 |
20 | 19 | rspcev 2955 | . . . . . . . 8 |
21 | 7, 13, 14, 20 | syl12anc 1180 | . . . . . . 7 |
22 | fveq2 5328 | . . . . . . . . . . . 12 | |
23 | 22 | eqeq2d 2364 | . . . . . . . . . . 11 |
24 | 23 | anbi1d 685 | . . . . . . . . . 10 |
25 | breq1 4642 | . . . . . . . . . 10 | |
26 | 24, 25 | anbi12d 691 | . . . . . . . . 9 |
27 | 26 | rexbidv 2635 | . . . . . . . 8 |
28 | 27 | rspcev 2955 | . . . . . . 7 |
29 | 4, 21, 28 | syl2anc 642 | . . . . . 6 |
30 | 29 | 3expib 1154 | . . . . 5 |
31 | simp3ll 1026 | . . . . . . . . 9 | |
32 | simp1 955 | . . . . . . . . . 10 | |
33 | simp2l 981 | . . . . . . . . . 10 | |
34 | f1of 5287 | . . . . . . . . . . 11 | |
35 | ffvelrn 5415 | . . . . . . . . . . 11 | |
36 | 34, 35 | sylan 457 | . . . . . . . . . 10 |
37 | 32, 33, 36 | syl2anc 642 | . . . . . . . . 9 |
38 | 31, 37 | eqeltrd 2427 | . . . . . . . 8 |
39 | simp3lr 1027 | . . . . . . . . 9 | |
40 | simp2r 982 | . . . . . . . . . 10 | |
41 | ffvelrn 5415 | . . . . . . . . . . 11 | |
42 | 34, 41 | sylan 457 | . . . . . . . . . 10 |
43 | 32, 40, 42 | syl2anc 642 | . . . . . . . . 9 |
44 | 39, 43 | eqeltrd 2427 | . . . . . . . 8 |
45 | simp3r 984 | . . . . . . . . 9 | |
46 | 31 | eqcomd 2358 | . . . . . . . . . 10 |
47 | f1ocnvfv 5478 | . . . . . . . . . . 11 | |
48 | 32, 33, 47 | syl2anc 642 | . . . . . . . . . 10 |
49 | 46, 48 | mpd 14 | . . . . . . . . 9 |
50 | 39 | eqcomd 2358 | . . . . . . . . . 10 |
51 | f1ocnvfv 5478 | . . . . . . . . . . 11 | |
52 | 32, 40, 51 | syl2anc 642 | . . . . . . . . . 10 |
53 | 50, 52 | mpd 14 | . . . . . . . . 9 |
54 | 45, 49, 53 | 3brtr4d 4669 | . . . . . . . 8 |
55 | 38, 44, 54 | jca31 520 | . . . . . . 7 |
56 | 55 | 3exp 1150 | . . . . . 6 |
57 | 56 | rexlimdvv 2744 | . . . . 5 |
58 | 30, 57 | impbid 183 | . . . 4 |
59 | 58 | opabbidv 4625 | . . 3 |
60 | 1, 59 | syl5eq 2397 | . 2 |
61 | f1oiso 5499 | . 2 | |
62 | 60, 61 | mpdan 649 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 w3a 934 wceq 1642 wcel 1710 wrex 2615 copab 4622 class class class wbr 4639 ccnv 4771 wf 4777 wf1o 4780 cfv 4781 wiso 4782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-iso 4796 |
This theorem is referenced by: (None) |
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