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Mirrors > Home > ILE Home > Th. List > isotr | Unicode version |
Description: Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
isotr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . . . 4 | |
2 | simpl 107 | . . . 4 | |
3 | f1oco 5169 | . . . 4 | |
4 | 1, 2, 3 | syl2anr 284 | . . 3 |
5 | f1of 5146 | . . . . . . . . . . . 12 | |
6 | 5 | ad2antrr 471 | . . . . . . . . . . 11 |
7 | simprl 497 | . . . . . . . . . . 11 | |
8 | 6, 7 | ffvelrnd 5324 | . . . . . . . . . 10 |
9 | simprr 498 | . . . . . . . . . . 11 | |
10 | 6, 9 | ffvelrnd 5324 | . . . . . . . . . 10 |
11 | simplrr 502 | . . . . . . . . . 10 | |
12 | breq1 3788 | . . . . . . . . . . . 12 | |
13 | fveq2 5198 | . . . . . . . . . . . . 13 | |
14 | 13 | breq1d 3795 | . . . . . . . . . . . 12 |
15 | 12, 14 | bibi12d 233 | . . . . . . . . . . 11 |
16 | breq2 3789 | . . . . . . . . . . . 12 | |
17 | fveq2 5198 | . . . . . . . . . . . . 13 | |
18 | 17 | breq2d 3797 | . . . . . . . . . . . 12 |
19 | 16, 18 | bibi12d 233 | . . . . . . . . . . 11 |
20 | 15, 19 | rspc2va 2714 | . . . . . . . . . 10 |
21 | 8, 10, 11, 20 | syl21anc 1168 | . . . . . . . . 9 |
22 | fvco3 5265 | . . . . . . . . . . 11 | |
23 | 6, 7, 22 | syl2anc 403 | . . . . . . . . . 10 |
24 | fvco3 5265 | . . . . . . . . . . 11 | |
25 | 6, 9, 24 | syl2anc 403 | . . . . . . . . . 10 |
26 | 23, 25 | breq12d 3798 | . . . . . . . . 9 |
27 | 21, 26 | bitr4d 189 | . . . . . . . 8 |
28 | 27 | bibi2d 230 | . . . . . . 7 |
29 | 28 | 2ralbidva 2388 | . . . . . 6 |
30 | 29 | biimpd 142 | . . . . 5 |
31 | 30 | impancom 256 | . . . 4 |
32 | 31 | imp 122 | . . 3 |
33 | 4, 32 | jca 300 | . 2 |
34 | df-isom 4931 | . . 3 | |
35 | df-isom 4931 | . . 3 | |
36 | 34, 35 | anbi12i 447 | . 2 |
37 | df-isom 4931 | . 2 | |
38 | 33, 36, 37 | 3imtr4i 199 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 class class class wbr 3785 ccom 4367 wf 4918 wf1o 4921 cfv 4922 wiso 4923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-isom 4931 |
This theorem is referenced by: (None) |
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