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Mirrors > Home > MPE Home > Th. List > wrdf | Structured version Visualization version GIF version |
Description: A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
wrdf | ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iswrd 14506 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
2 | simpr 483 | . . . 4 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^𝑙)⟶𝑆) | |
3 | fnfzo0hash 14449 | . . . . . 6 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (♯‘𝑊) = 𝑙) | |
4 | 3 | oveq2d 7442 | . . . . 5 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (0..^(♯‘𝑊)) = (0..^𝑙)) |
5 | 4 | feq2d 6713 | . . . 4 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (𝑊:(0..^(♯‘𝑊))⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆)) |
6 | 2, 5 | mpbird 256 | . . 3 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
7 | 6 | rexlimiva 3144 | . 2 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ∃wrex 3067 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 0cc0 11146 ℕ0cn0 12510 ..^cfzo 13667 ♯chash 14329 Word cword 14504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 |
This theorem is referenced by: iswrdb 14510 wrddm 14511 wrdsymbcl 14517 wrdfn 14518 wrdffz 14525 0wrd0 14530 wrdsymb 14532 wrdnval 14535 wrdred1 14550 wrdred1hash 14551 ccatcl 14564 ccatalpha 14583 s1dm 14598 swrdcl 14635 swrdf 14640 swrdwrdsymb 14652 pfxres 14669 cats1un 14711 revcl 14751 revlen 14752 revrev 14757 repsdf2 14768 cshwf 14790 cshinj 14801 wrdco 14822 lenco 14823 revco 14825 ccatco 14826 lswco 14830 s2dm 14881 wwlktovf 14947 ofccat 14956 gsumwsubmcl 18796 gsumsgrpccat 18799 gsumwmhm 18804 frmdss2 18822 symgtrinv 19434 psgnunilem5 19456 psgnunilem2 19457 psgnunilem3 19458 efginvrel1 19690 efgsf 19691 efgsrel 19696 efgs1b 19698 efgredlemf 19703 efgredlemd 19706 efgredlemc 19707 efgredlem 19709 frgpup3lem 19739 pgpfaclem1 20045 ablfaclem2 20050 ablfaclem3 20051 ablfac2 20053 dchrptlem1 27217 dchrptlem2 27218 trgcgrg 28339 tgcgr4 28355 wrdupgr 28918 wrdumgr 28930 vdegp1ai 29370 vdegp1bi 29371 wlkres 29504 wlkp1 29515 wlkdlem1 29516 trlf1 29532 trlreslem 29533 upgrwlkdvdelem 29570 pthdlem1 29600 pthdlem2lem 29601 uspgrn2crct 29639 wlkiswwlks2lem3 29702 wlkiswwlksupgr2 29708 clwlkclwwlklem2a 29828 clwlkclwwlklem2 29830 1wlkdlem1 29967 wlk2v2e 29987 eucrctshift 30073 konigsbergssiedgw 30080 wrdfd 32680 wrdres 32681 pfxf1 32686 s3f1 32691 ccatf1 32693 swrdrn3 32697 cycpmcl 32858 tocyc01 32860 cycpmco2rn 32867 cycpmrn 32885 tocyccntz 32886 cycpmconjslem2 32897 sseqf 34045 fiblem 34051 ofcccat 34208 signstcl 34230 signstf 34231 signstfvn 34234 signsvtn0 34235 signstres 34240 signsvtp 34248 signsvtn 34249 signsvfpn 34250 signsvfnn 34251 signshf 34253 revwlk 34767 mvrsfpw 35149 frlmfzowrdb 41775 amgm2d 43659 amgm3d 43660 amgm4d 43661 lswn0 46813 amgmw2d 48315 |
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