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Mirrors > Home > ILE Home > Th. List > isocnv | Unicode version |
Description: Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Ref | Expression |
---|---|
isocnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5139 | . . . 4 | |
2 | 1 | adantr 261 | . . 3 |
3 | f1ocnvfv2 5418 | . . . . . . . 8 | |
4 | 3 | adantrr 448 | . . . . . . 7 |
5 | f1ocnvfv2 5418 | . . . . . . . 8 | |
6 | 5 | adantrl 447 | . . . . . . 7 |
7 | 4, 6 | breq12d 3777 | . . . . . 6 |
8 | 7 | adantlr 446 | . . . . 5 |
9 | f1of 5126 | . . . . . . 7 | |
10 | 1, 9 | syl 14 | . . . . . 6 |
11 | ffvelrn 5300 | . . . . . . . . 9 | |
12 | ffvelrn 5300 | . . . . . . . . 9 | |
13 | 11, 12 | anim12dan 532 | . . . . . . . 8 |
14 | breq1 3767 | . . . . . . . . . . 11 | |
15 | fveq2 5178 | . . . . . . . . . . . 12 | |
16 | 15 | breq1d 3774 | . . . . . . . . . . 11 |
17 | 14, 16 | bibi12d 224 | . . . . . . . . . 10 |
18 | bicom 128 | . . . . . . . . . 10 | |
19 | 17, 18 | syl6bb 185 | . . . . . . . . 9 |
20 | fveq2 5178 | . . . . . . . . . . 11 | |
21 | 20 | breq2d 3776 | . . . . . . . . . 10 |
22 | breq2 3768 | . . . . . . . . . 10 | |
23 | 21, 22 | bibi12d 224 | . . . . . . . . 9 |
24 | 19, 23 | rspc2va 2663 | . . . . . . . 8 |
25 | 13, 24 | sylan 267 | . . . . . . 7 |
26 | 25 | an32s 502 | . . . . . 6 |
27 | 10, 26 | sylanl1 382 | . . . . 5 |
28 | 8, 27 | bitr3d 179 | . . . 4 |
29 | 28 | ralrimivva 2401 | . . 3 |
30 | 2, 29 | jca 290 | . 2 |
31 | df-isom 4911 | . 2 | |
32 | df-isom 4911 | . 2 | |
33 | 30, 31, 32 | 3imtr4i 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 class class class wbr 3764 ccnv 4344 wf 4898 wf1o 4901 cfv 4902 wiso 4903 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-isom 4911 |
This theorem is referenced by: isores1 5454 isose 5460 isopo 5462 isoso 5464 |
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