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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 108 | . . . 4 | |
2 | simpl 108 | . . . 4 | |
3 | f1oco 5465 | . . . 4 | |
4 | 1, 2, 3 | syl2anr 288 | . . 3 |
5 | f1of 5442 | . . . . . . . . . . . 12 | |
6 | 5 | ad2antrr 485 | . . . . . . . . . . 11 |
7 | simprl 526 | . . . . . . . . . . 11 | |
8 | 6, 7 | ffvelrnd 5632 | . . . . . . . . . 10 |
9 | simprr 527 | . . . . . . . . . . 11 | |
10 | 6, 9 | ffvelrnd 5632 | . . . . . . . . . 10 |
11 | simplrr 531 | . . . . . . . . . 10 | |
12 | breq1 3992 | . . . . . . . . . . . 12 | |
13 | fveq2 5496 | . . . . . . . . . . . . 13 | |
14 | 13 | breq1d 3999 | . . . . . . . . . . . 12 |
15 | 12, 14 | bibi12d 234 | . . . . . . . . . . 11 |
16 | breq2 3993 | . . . . . . . . . . . 12 | |
17 | fveq2 5496 | . . . . . . . . . . . . 13 | |
18 | 17 | breq2d 4001 | . . . . . . . . . . . 12 |
19 | 16, 18 | bibi12d 234 | . . . . . . . . . . 11 |
20 | 15, 19 | rspc2va 2848 | . . . . . . . . . 10 |
21 | 8, 10, 11, 20 | syl21anc 1232 | . . . . . . . . 9 |
22 | fvco3 5567 | . . . . . . . . . . 11 | |
23 | 6, 7, 22 | syl2anc 409 | . . . . . . . . . 10 |
24 | fvco3 5567 | . . . . . . . . . . 11 | |
25 | 6, 9, 24 | syl2anc 409 | . . . . . . . . . 10 |
26 | 23, 25 | breq12d 4002 | . . . . . . . . 9 |
27 | 21, 26 | bitr4d 190 | . . . . . . . 8 |
28 | 27 | bibi2d 231 | . . . . . . 7 |
29 | 28 | 2ralbidva 2492 | . . . . . 6 |
30 | 29 | biimpd 143 | . . . . 5 |
31 | 30 | impancom 258 | . . . 4 |
32 | 31 | imp 123 | . . 3 |
33 | 4, 32 | jca 304 | . 2 |
34 | df-isom 5207 | . . 3 | |
35 | df-isom 5207 | . . 3 | |
36 | 34, 35 | anbi12i 457 | . 2 |
37 | df-isom 5207 | . 2 | |
38 | 33, 36, 37 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 class class class wbr 3989 ccom 4615 wf 5194 wf1o 5197 cfv 5198 wiso 5199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 |
This theorem is referenced by: (None) |
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