Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5390 | . . . 4 | |
4 | 1, 2, 3 | syl2anr 288 | . . 3 |
5 | f1of 5367 | . . . . . . . . . . . 12 | |
6 | 5 | ad2antrr 479 | . . . . . . . . . . 11 |
7 | simprl 520 | . . . . . . . . . . 11 | |
8 | 6, 7 | ffvelrnd 5556 | . . . . . . . . . 10 |
9 | simprr 521 | . . . . . . . . . . 11 | |
10 | 6, 9 | ffvelrnd 5556 | . . . . . . . . . 10 |
11 | simplrr 525 | . . . . . . . . . 10 | |
12 | breq1 3932 | . . . . . . . . . . . 12 | |
13 | fveq2 5421 | . . . . . . . . . . . . 13 | |
14 | 13 | breq1d 3939 | . . . . . . . . . . . 12 |
15 | 12, 14 | bibi12d 234 | . . . . . . . . . . 11 |
16 | breq2 3933 | . . . . . . . . . . . 12 | |
17 | fveq2 5421 | . . . . . . . . . . . . 13 | |
18 | 17 | breq2d 3941 | . . . . . . . . . . . 12 |
19 | 16, 18 | bibi12d 234 | . . . . . . . . . . 11 |
20 | 15, 19 | rspc2va 2803 | . . . . . . . . . 10 |
21 | 8, 10, 11, 20 | syl21anc 1215 | . . . . . . . . 9 |
22 | fvco3 5492 | . . . . . . . . . . 11 | |
23 | 6, 7, 22 | syl2anc 408 | . . . . . . . . . 10 |
24 | fvco3 5492 | . . . . . . . . . . 11 | |
25 | 6, 9, 24 | syl2anc 408 | . . . . . . . . . 10 |
26 | 23, 25 | breq12d 3942 | . . . . . . . . 9 |
27 | 21, 26 | bitr4d 190 | . . . . . . . 8 |
28 | 27 | bibi2d 231 | . . . . . . 7 |
29 | 28 | 2ralbidva 2457 | . . . . . 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 5132 | . . 3 | |
35 | df-isom 5132 | . . 3 | |
36 | 34, 35 | anbi12i 455 | . 2 |
37 | df-isom 5132 | . 2 | |
38 | 33, 36, 37 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 class class class wbr 3929 ccom 4543 wf 5119 wf1o 5122 cfv 5123 wiso 5124 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |