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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 13576 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
2 | simpr 479 | . . . 4 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^𝑙)⟶𝑆) | |
3 | fnfzo0hash 13523 | . . . . . 6 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (♯‘𝑊) = 𝑙) | |
4 | 3 | oveq2d 6921 | . . . . 5 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (0..^(♯‘𝑊)) = (0..^𝑙)) |
5 | 4 | feq2d 6264 | . . . 4 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (𝑊:(0..^(♯‘𝑊))⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆)) |
6 | 2, 5 | mpbird 249 | . . 3 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
7 | 6 | rexlimiva 3237 | . 2 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
8 | 1, 7 | sylbi 209 | 1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ∃wrex 3118 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 0cc0 10252 ℕ0cn0 11618 ..^cfzo 12760 ♯chash 13410 Word cword 13574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 |
This theorem is referenced by: iswrdb 13580 wrddm 13581 wrdsymbcl 13587 wrdfn 13588 wrdv 13589 wrdffz 13595 0wrd0 13600 wrdsymb 13602 wrdnval 13605 wrdred1 13620 wrdred1hash 13621 ccatcl 13634 ccatass 13648 ccatrn 13649 ccatalpha 13653 s1dm 13668 swrdcl 13705 swrd0valOLD 13707 swrdf 13712 swrdnd2 13720 swrdwrdsymb 13736 ccatswrd 13746 swrdccat1OLD 13747 swrdccat2 13748 pfxres 13758 ccatpfx 13780 cats1un 13811 revcl 13877 revlen 13878 revccat 13882 revrev 13883 repsdf2 13894 cshwf 13921 cshinj 13932 wrdco 13952 lenco 13953 revco 13955 ccatco 13956 lswco 13960 s2dm 14011 wwlktovf 14078 ofccat 14087 gsumwsubmcl 17728 gsumccat 17731 gsumwmhm 17736 frmdss2 17754 symgtrinv 18242 psgnunilem5 18264 psgnunilem5OLD 18265 psgnunilem2 18266 psgnunilem3 18267 efginvrel1 18492 efgsf 18493 efgsrel 18498 efgs1b 18500 efgredlemf 18506 efgredlemd 18509 efgredlemc 18510 efgredlem 18512 efgredlemOLD 18513 frgpup3lem 18543 pgpfaclem1 18834 ablfaclem2 18839 ablfaclem3 18840 ablfac2 18842 dchrptlem1 25402 dchrptlem2 25403 trgcgrg 25827 tgcgr4 25843 wrdupgr 26383 wrdumgr 26395 vdegp1ai 26834 vdegp1bi 26835 wlkres 26969 wlkreslemOLD 26970 wlkresOLD 26971 wlkp1 26982 wlkdlem1 26983 trlf1 26999 trlreslem 27000 trlreslemOLD 27002 upgrwlkdvdelem 27038 pthdlem1 27068 pthdlem2lem 27069 uspgrn2crct 27107 wlkiswwlks2lem3 27170 wlkiswwlksupgr2 27176 clwlkclwwlklem2a 27327 clwlkclwwlklem2 27329 1wlkdlem1 27513 wlk2v2e 27533 eucrctshift 27620 konigsbergssiedgw 27629 sseqf 31000 fiblem 31006 wrdfd 31162 wrdres 31163 ofcccat 31167 signstcl 31189 signstf 31190 signstfvn 31193 signsvtn0 31194 signsvtn0OLD 31195 signstres 31200 signsvtp 31209 signsvtn 31210 signsvfpn 31211 signsvfnn 31212 signshf 31214 mvrsfpw 31949 amgm2d 39341 amgm3d 39342 amgm4d 39343 lswn0 42268 amgmw2d 43446 |
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