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 5466 | . . . 4 | |
2 | 1 | adantr 276 | . . 3 |
3 | f1ocnvfv2 5769 | . . . . . . . 8 | |
4 | 3 | adantrr 479 | . . . . . . 7 |
5 | f1ocnvfv2 5769 | . . . . . . . 8 | |
6 | 5 | adantrl 478 | . . . . . . 7 |
7 | 4, 6 | breq12d 4011 | . . . . . 6 |
8 | 7 | adantlr 477 | . . . . 5 |
9 | f1of 5453 | . . . . . . 7 | |
10 | 1, 9 | syl 14 | . . . . . 6 |
11 | ffvelcdm 5641 | . . . . . . . . 9 | |
12 | ffvelcdm 5641 | . . . . . . . . 9 | |
13 | 11, 12 | anim12dan 600 | . . . . . . . 8 |
14 | breq1 4001 | . . . . . . . . . . 11 | |
15 | fveq2 5507 | . . . . . . . . . . . 12 | |
16 | 15 | breq1d 4008 | . . . . . . . . . . 11 |
17 | 14, 16 | bibi12d 235 | . . . . . . . . . 10 |
18 | bicom 140 | . . . . . . . . . 10 | |
19 | 17, 18 | bitrdi 196 | . . . . . . . . 9 |
20 | fveq2 5507 | . . . . . . . . . . 11 | |
21 | 20 | breq2d 4010 | . . . . . . . . . 10 |
22 | breq2 4002 | . . . . . . . . . 10 | |
23 | 21, 22 | bibi12d 235 | . . . . . . . . 9 |
24 | 19, 23 | rspc2va 2853 | . . . . . . . 8 |
25 | 13, 24 | sylan 283 | . . . . . . 7 |
26 | 25 | an32s 568 | . . . . . 6 |
27 | 10, 26 | sylanl1 402 | . . . . 5 |
28 | 8, 27 | bitr3d 190 | . . . 4 |
29 | 28 | ralrimivva 2557 | . . 3 |
30 | 2, 29 | jca 306 | . 2 |
31 | df-isom 5217 | . 2 | |
32 | df-isom 5217 | . 2 | |
33 | 30, 31, 32 | 3imtr4i 201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 class class class wbr 3998 ccnv 4619 wf 5204 wf1o 5207 cfv 5208 wiso 5209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 |
This theorem is referenced by: isores1 5805 isose 5812 isopo 5814 isoso 5816 isoti 6996 infrenegsupex 9567 infxrnegsupex 11239 relogiso 13874 |
Copyright terms: Public domain | W3C validator |