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Mirrors > Home > NFE Home > Th. List > isocnv | Unicode version |
Description: Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 27-Apr-2004.) |
Ref | Expression |
---|---|
isocnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5299 | . . . 4 | |
2 | 1 | adantr 451 | . . 3 |
3 | f1ocnvfv2 5477 | . . . . . . . 8 | |
4 | 3 | adantrr 697 | . . . . . . 7 |
5 | f1ocnvfv2 5477 | . . . . . . . 8 | |
6 | 5 | adantrl 696 | . . . . . . 7 |
7 | 4, 6 | breq12d 4652 | . . . . . 6 |
8 | 7 | adantlr 695 | . . . . 5 |
9 | f1of 5287 | . . . . . . 7 | |
10 | 1, 9 | syl 15 | . . . . . 6 |
11 | ffvelrn 5415 | . . . . . . . . 9 | |
12 | ffvelrn 5415 | . . . . . . . . 9 | |
13 | 11, 12 | anim12dan 810 | . . . . . . . 8 |
14 | breq1 4642 | . . . . . . . . . . 11 | |
15 | fveq2 5328 | . . . . . . . . . . . 12 | |
16 | 15 | breq1d 4649 | . . . . . . . . . . 11 |
17 | 14, 16 | bibi12d 312 | . . . . . . . . . 10 |
18 | bicom 191 | . . . . . . . . . 10 | |
19 | 17, 18 | syl6bb 252 | . . . . . . . . 9 |
20 | fveq2 5328 | . . . . . . . . . . 11 | |
21 | 20 | breq2d 4651 | . . . . . . . . . 10 |
22 | breq2 4643 | . . . . . . . . . 10 | |
23 | 21, 22 | bibi12d 312 | . . . . . . . . 9 |
24 | 19, 23 | rspc2va 2962 | . . . . . . . 8 |
25 | 13, 24 | sylan 457 | . . . . . . 7 |
26 | 25 | an32s 779 | . . . . . 6 |
27 | 10, 26 | sylanl1 631 | . . . . 5 |
28 | 8, 27 | bitr3d 246 | . . . 4 |
29 | 28 | ralrimivva 2706 | . . 3 |
30 | 2, 29 | jca 518 | . 2 |
31 | df-iso 4796 | . 2 | |
32 | df-iso 4796 | . 2 | |
33 | 30, 31, 32 | 3imtr4i 257 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wral 2614 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: isores1 5494 |
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