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Mirrors > Home > NFE Home > Th. List > f1oiso | Unicode version |
Description: Any one-to-one onto function determines an isomorphism with an induced relation . Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by set.mm contributors, 30-Apr-2004.) |
Ref | Expression |
---|---|
f1oiso |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 443 | . 2 | |
2 | f1of1 5286 | . . 3 | |
3 | df-br 4640 | . . . . 5 | |
4 | eleq2 2414 | . . . . . . 7 | |
5 | fvex 5339 | . . . . . . . . 9 | |
6 | fvex 5339 | . . . . . . . . 9 | |
7 | eqeq1 2359 | . . . . . . . . . . . 12 | |
8 | 7 | anbi1d 685 | . . . . . . . . . . 11 |
9 | 8 | anbi1d 685 | . . . . . . . . . 10 |
10 | 9 | 2rexbidv 2657 | . . . . . . . . 9 |
11 | eqeq1 2359 | . . . . . . . . . . . 12 | |
12 | 11 | anbi2d 684 | . . . . . . . . . . 11 |
13 | 12 | anbi1d 685 | . . . . . . . . . 10 |
14 | 13 | 2rexbidv 2657 | . . . . . . . . 9 |
15 | 5, 6, 10, 14 | opelopab 4708 | . . . . . . . 8 |
16 | anass 630 | . . . . . . . . . . . . . . 15 | |
17 | f1fveq 5473 | . . . . . . . . . . . . . . . . . 18 | |
18 | eqcom 2355 | . . . . . . . . . . . . . . . . . 18 | |
19 | 17, 18 | syl6bb 252 | . . . . . . . . . . . . . . . . 17 |
20 | 19 | anassrs 629 | . . . . . . . . . . . . . . . 16 |
21 | 20 | anbi1d 685 | . . . . . . . . . . . . . . 15 |
22 | 16, 21 | syl5bb 248 | . . . . . . . . . . . . . 14 |
23 | 22 | rexbidv 2635 | . . . . . . . . . . . . 13 |
24 | r19.42v 2765 | . . . . . . . . . . . . 13 | |
25 | 23, 24 | syl6bb 252 | . . . . . . . . . . . 12 |
26 | 25 | rexbidva 2631 | . . . . . . . . . . 11 |
27 | breq1 4642 | . . . . . . . . . . . . . . 15 | |
28 | 27 | anbi2d 684 | . . . . . . . . . . . . . 14 |
29 | 28 | rexbidv 2635 | . . . . . . . . . . . . 13 |
30 | 29 | ceqsrexv 2972 | . . . . . . . . . . . 12 |
31 | 30 | adantl 452 | . . . . . . . . . . 11 |
32 | 26, 31 | bitrd 244 | . . . . . . . . . 10 |
33 | f1fveq 5473 | . . . . . . . . . . . . . . 15 | |
34 | eqcom 2355 | . . . . . . . . . . . . . . 15 | |
35 | 33, 34 | syl6bb 252 | . . . . . . . . . . . . . 14 |
36 | 35 | anassrs 629 | . . . . . . . . . . . . 13 |
37 | 36 | anbi1d 685 | . . . . . . . . . . . 12 |
38 | 37 | rexbidva 2631 | . . . . . . . . . . 11 |
39 | breq2 4643 | . . . . . . . . . . . . 13 | |
40 | 39 | ceqsrexv 2972 | . . . . . . . . . . . 12 |
41 | 40 | adantl 452 | . . . . . . . . . . 11 |
42 | 38, 41 | bitrd 244 | . . . . . . . . . 10 |
43 | 32, 42 | sylan9bb 680 | . . . . . . . . 9 |
44 | 43 | anandis 803 | . . . . . . . 8 |
45 | 15, 44 | syl5bb 248 | . . . . . . 7 |
46 | 4, 45 | sylan9bbr 681 | . . . . . 6 |
47 | 46 | an32s 779 | . . . . 5 |
48 | 3, 47 | syl5rbb 249 | . . . 4 |
49 | 48 | ralrimivva 2706 | . . 3 |
50 | 2, 49 | sylan 457 | . 2 |
51 | df-iso 4796 | . 2 | |
52 | 1, 50, 51 | sylanbrc 645 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wral 2614 wrex 2615 cop 4561 copab 4622 class class class wbr 4639 wf1 4778 wf1o 4780 cfv 4781 wiso 4782 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-f1o 4794 df-fv 4795 df-iso 4796 |
This theorem is referenced by: f1oiso2 5500 |
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