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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 5682 | . . . . . . . . 9 | |
3 | 2 | adantrr 470 | . . . . . . . 8 |
4 | 3 | 3adant3 1001 | . . . . . . 7 |
5 | f1ocnvdm 5682 | . . . . . . . . . 10 | |
6 | 5 | adantrl 469 | . . . . . . . . 9 |
7 | 6 | 3adant3 1001 | . . . . . . . 8 |
8 | f1ocnvfv2 5679 | . . . . . . . . . . 11 | |
9 | 8 | eqcomd 2145 | . . . . . . . . . 10 |
10 | f1ocnvfv2 5679 | . . . . . . . . . . 11 | |
11 | 10 | eqcomd 2145 | . . . . . . . . . 10 |
12 | 9, 11 | anim12dan 589 | . . . . . . . . 9 |
13 | 12 | 3adant3 1001 | . . . . . . . 8 |
14 | simp3 983 | . . . . . . . 8 | |
15 | fveq2 5421 | . . . . . . . . . . . 12 | |
16 | 15 | eqeq2d 2151 | . . . . . . . . . . 11 |
17 | 16 | anbi2d 459 | . . . . . . . . . 10 |
18 | breq2 3933 | . . . . . . . . . 10 | |
19 | 17, 18 | anbi12d 464 | . . . . . . . . 9 |
20 | 19 | rspcev 2789 | . . . . . . . 8 |
21 | 7, 13, 14, 20 | syl12anc 1214 | . . . . . . 7 |
22 | fveq2 5421 | . . . . . . . . . . . 12 | |
23 | 22 | eqeq2d 2151 | . . . . . . . . . . 11 |
24 | 23 | anbi1d 460 | . . . . . . . . . 10 |
25 | breq1 3932 | . . . . . . . . . 10 | |
26 | 24, 25 | anbi12d 464 | . . . . . . . . 9 |
27 | 26 | rexbidv 2438 | . . . . . . . 8 |
28 | 27 | rspcev 2789 | . . . . . . 7 |
29 | 4, 21, 28 | syl2anc 408 | . . . . . 6 |
30 | 29 | 3expib 1184 | . . . . 5 |
31 | simp3ll 1052 | . . . . . . . . 9 | |
32 | simp1 981 | . . . . . . . . . 10 | |
33 | simp2l 1007 | . . . . . . . . . 10 | |
34 | f1of 5367 | . . . . . . . . . . 11 | |
35 | 34 | ffvelrnda 5555 | . . . . . . . . . 10 |
36 | 32, 33, 35 | syl2anc 408 | . . . . . . . . 9 |
37 | 31, 36 | eqeltrd 2216 | . . . . . . . 8 |
38 | simp3lr 1053 | . . . . . . . . 9 | |
39 | simp2r 1008 | . . . . . . . . . 10 | |
40 | 34 | ffvelrnda 5555 | . . . . . . . . . 10 |
41 | 32, 39, 40 | syl2anc 408 | . . . . . . . . 9 |
42 | 38, 41 | eqeltrd 2216 | . . . . . . . 8 |
43 | simp3r 1010 | . . . . . . . . 9 | |
44 | 31 | eqcomd 2145 | . . . . . . . . . 10 |
45 | f1ocnvfv 5680 | . . . . . . . . . . 11 | |
46 | 32, 33, 45 | syl2anc 408 | . . . . . . . . . 10 |
47 | 44, 46 | mpd 13 | . . . . . . . . 9 |
48 | 38 | eqcomd 2145 | . . . . . . . . . 10 |
49 | f1ocnvfv 5680 | . . . . . . . . . . 11 | |
50 | 32, 39, 49 | syl2anc 408 | . . . . . . . . . 10 |
51 | 48, 50 | mpd 13 | . . . . . . . . 9 |
52 | 43, 47, 51 | 3brtr4d 3960 | . . . . . . . 8 |
53 | 37, 42, 52 | jca31 307 | . . . . . . 7 |
54 | 53 | 3exp 1180 | . . . . . 6 |
55 | 54 | rexlimdvv 2556 | . . . . 5 |
56 | 30, 55 | impbid 128 | . . . 4 |
57 | 56 | opabbidv 3994 | . . 3 |
58 | 1, 57 | syl5eq 2184 | . 2 |
59 | f1oiso 5727 | . 2 | |
60 | 58, 59 | mpdan 417 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929 copab 3988 ccnv 4538 wf1o 5122 cfv 5123 wiso 5124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 |
This theorem is referenced by: (None) |
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