Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > shftfn | Unicode version |
Description: Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 |
Ref | Expression |
---|---|
shftfn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4482 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | fnfun 5016 | . . . . . 6 | |
4 | 3 | adantr 270 | . . . . 5 |
5 | funmo 4937 | . . . . . . 7 | |
6 | vex 2604 | . . . . . . . . . 10 | |
7 | vex 2604 | . . . . . . . . . 10 | |
8 | eleq1 2141 | . . . . . . . . . . 11 | |
9 | oveq1 5539 | . . . . . . . . . . . 12 | |
10 | 9 | breq1d 3795 | . . . . . . . . . . 11 |
11 | 8, 10 | anbi12d 456 | . . . . . . . . . 10 |
12 | breq2 3789 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 451 | . . . . . . . . . 10 |
14 | eqid 2081 | . . . . . . . . . 10 | |
15 | 6, 7, 11, 13, 14 | brab 4027 | . . . . . . . . 9 |
16 | 15 | simprbi 269 | . . . . . . . 8 |
17 | 16 | moimi 2006 | . . . . . . 7 |
18 | 5, 17 | syl 14 | . . . . . 6 |
19 | 18 | alrimiv 1795 | . . . . 5 |
20 | 4, 19 | syl 14 | . . . 4 |
21 | dffun6 4936 | . . . 4 | |
22 | 2, 20, 21 | sylanbrc 408 | . . 3 |
23 | shftfval.1 | . . . . . 6 | |
24 | 23 | shftfval 9709 | . . . . 5 |
25 | 24 | adantl 271 | . . . 4 |
26 | 25 | funeqd 4943 | . . 3 |
27 | 22, 26 | mpbird 165 | . 2 |
28 | 23 | shftdm 9710 | . . 3 |
29 | fndm 5018 | . . . . 5 | |
30 | 29 | eleq2d 2148 | . . . 4 |
31 | 30 | rabbidv 2593 | . . 3 |
32 | 28, 31 | sylan9eqr 2135 | . 2 |
33 | df-fn 4925 | . 2 | |
34 | 27, 32, 33 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wcel 1433 wmo 1942 crab 2352 cvv 2601 class class class wbr 3785 copab 3838 cdm 4363 wrel 4368 wfun 4916 wfn 4917 (class class class)co 5532 cc 6979 cmin 7279 cshi 9702 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-shft 9703 |
This theorem is referenced by: shftf 9718 |
Copyright terms: Public domain | W3C validator |