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Mirrors > Home > MPE Home > Th. List > isotr | Structured version Visualization version 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 473 | . . . 4 | |
2 | simpl 473 | . . . 4 | |
3 | f1oco 6159 | . . . 4 | |
4 | 1, 2, 3 | syl2anr 495 | . . 3 |
5 | f1of 6137 | . . . . . . . . . . . 12 | |
6 | 5 | ad2antrr 762 | . . . . . . . . . . 11 |
7 | simprl 794 | . . . . . . . . . . 11 | |
8 | 6, 7 | ffvelrnd 6360 | . . . . . . . . . 10 |
9 | simprr 796 | . . . . . . . . . . 11 | |
10 | 6, 9 | ffvelrnd 6360 | . . . . . . . . . 10 |
11 | simplrr 801 | . . . . . . . . . 10 | |
12 | breq1 4656 | . . . . . . . . . . . 12 | |
13 | fveq2 6191 | . . . . . . . . . . . . 13 | |
14 | 13 | breq1d 4663 | . . . . . . . . . . . 12 |
15 | 12, 14 | bibi12d 335 | . . . . . . . . . . 11 |
16 | breq2 4657 | . . . . . . . . . . . 12 | |
17 | fveq2 6191 | . . . . . . . . . . . . 13 | |
18 | 17 | breq2d 4665 | . . . . . . . . . . . 12 |
19 | 16, 18 | bibi12d 335 | . . . . . . . . . . 11 |
20 | 15, 19 | rspc2va 3323 | . . . . . . . . . 10 |
21 | 8, 10, 11, 20 | syl21anc 1325 | . . . . . . . . 9 |
22 | fvco3 6275 | . . . . . . . . . . 11 | |
23 | 6, 7, 22 | syl2anc 693 | . . . . . . . . . 10 |
24 | fvco3 6275 | . . . . . . . . . . 11 | |
25 | 6, 9, 24 | syl2anc 693 | . . . . . . . . . 10 |
26 | 23, 25 | breq12d 4666 | . . . . . . . . 9 |
27 | 21, 26 | bitr4d 271 | . . . . . . . 8 |
28 | 27 | bibi2d 332 | . . . . . . 7 |
29 | 28 | 2ralbidva 2988 | . . . . . 6 |
30 | 29 | biimpd 219 | . . . . 5 |
31 | 30 | impancom 456 | . . . 4 |
32 | 31 | imp 445 | . . 3 |
33 | 4, 32 | jca 554 | . 2 |
34 | df-isom 5897 | . . 3 | |
35 | df-isom 5897 | . . 3 | |
36 | 34, 35 | anbi12i 733 | . 2 |
37 | df-isom 5897 | . 2 | |
38 | 33, 36, 37 | 3imtr4i 281 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 class class class wbr 4653 ccom 5118 wf 5884 wf1o 5887 cfv 5888 wiso 5889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 |
This theorem is referenced by: weisoeq 6605 oieu 8444 fz1isolem 13245 erdsze2lem2 31186 fzisoeu 39514 |
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