Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5439 | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | f1ocnvfv2 5740 | . . . . . . . 8 | |
4 | 3 | adantrr 471 | . . . . . . 7 |
5 | f1ocnvfv2 5740 | . . . . . . . 8 | |
6 | 5 | adantrl 470 | . . . . . . 7 |
7 | 4, 6 | breq12d 3989 | . . . . . 6 |
8 | 7 | adantlr 469 | . . . . 5 |
9 | f1of 5426 | . . . . . . 7 | |
10 | 1, 9 | syl 14 | . . . . . 6 |
11 | ffvelrn 5612 | . . . . . . . . 9 | |
12 | ffvelrn 5612 | . . . . . . . . 9 | |
13 | 11, 12 | anim12dan 590 | . . . . . . . 8 |
14 | breq1 3979 | . . . . . . . . . . 11 | |
15 | fveq2 5480 | . . . . . . . . . . . 12 | |
16 | 15 | breq1d 3986 | . . . . . . . . . . 11 |
17 | 14, 16 | bibi12d 234 | . . . . . . . . . 10 |
18 | bicom 139 | . . . . . . . . . 10 | |
19 | 17, 18 | bitrdi 195 | . . . . . . . . 9 |
20 | fveq2 5480 | . . . . . . . . . . 11 | |
21 | 20 | breq2d 3988 | . . . . . . . . . 10 |
22 | breq2 3980 | . . . . . . . . . 10 | |
23 | 21, 22 | bibi12d 234 | . . . . . . . . 9 |
24 | 19, 23 | rspc2va 2839 | . . . . . . . 8 |
25 | 13, 24 | sylan 281 | . . . . . . 7 |
26 | 25 | an32s 558 | . . . . . 6 |
27 | 10, 26 | sylanl1 400 | . . . . 5 |
28 | 8, 27 | bitr3d 189 | . . . 4 |
29 | 28 | ralrimivva 2546 | . . 3 |
30 | 2, 29 | jca 304 | . 2 |
31 | df-isom 5191 | . 2 | |
32 | df-isom 5191 | . 2 | |
33 | 30, 31, 32 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 class class class wbr 3976 ccnv 4597 wf 5178 wf1o 5181 cfv 5182 wiso 5183 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 |
This theorem is referenced by: isores1 5776 isose 5783 isopo 5785 isoso 5787 isoti 6963 infrenegsupex 9523 infxrnegsupex 11190 relogiso 13341 |
Copyright terms: Public domain | W3C validator |