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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 14554 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) | |
| 2 | simpr 484 | . . . 4 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^𝑙)⟶𝑆) | |
| 3 | fnfzo0hash 14489 | . . . . . 6 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (♯‘𝑊) = 𝑙) | |
| 4 | 3 | oveq2d 7447 | . . . . 5 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (0..^(♯‘𝑊)) = (0..^𝑙)) |
| 5 | 4 | feq2d 6722 | . . . 4 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → (𝑊:(0..^(♯‘𝑊))⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆)) |
| 6 | 2, 5 | mpbird 257 | . . 3 ⊢ ((𝑙 ∈ ℕ0 ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
| 7 | 6 | rexlimiva 3147 | . 2 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
| 8 | 1, 7 | sylbi 217 | 1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕ0cn0 12526 ..^cfzo 13694 ♯chash 14369 Word cword 14552 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 |
| This theorem is referenced by: iswrdb 14558 wrddm 14559 wrdsymbcl 14565 wrdfn 14566 wrdffz 14573 0wrd0 14578 wrdsymb 14580 wrdnval 14583 wrdred1 14598 wrdred1hash 14599 ccatcl 14612 ccatalpha 14631 s1dm 14646 swrdcl 14683 swrdf 14688 swrdwrdsymb 14700 pfxres 14717 cats1un 14759 revcl 14799 revlen 14800 revrev 14805 repsdf2 14816 cshwf 14838 cshinj 14849 wrdco 14870 lenco 14871 revco 14873 ccatco 14874 lswco 14878 s2dm 14929 wwlktovf 14995 s7f1o 15005 ofccat 15008 gsumwsubmcl 18850 gsumsgrpccat 18853 gsumwmhm 18858 frmdss2 18876 symgtrinv 19490 psgnunilem5 19512 psgnunilem2 19513 psgnunilem3 19514 efginvrel1 19746 efgsf 19747 efgsrel 19752 efgs1b 19754 efgredlemf 19759 efgredlemd 19762 efgredlemc 19763 efgredlem 19765 frgpup3lem 19795 pgpfaclem1 20101 ablfaclem2 20106 ablfaclem3 20107 ablfac2 20109 dchrptlem1 27308 dchrptlem2 27309 trgcgrg 28523 tgcgr4 28539 wrdupgr 29102 wrdumgr 29114 vdegp1ai 29554 vdegp1bi 29555 wlkres 29688 wlkp1 29699 wlkdlem1 29700 trlf1 29716 trlreslem 29717 upgrwlkdvdelem 29756 pthdlem1 29786 pthdlem2lem 29787 uspgrn2crct 29828 wlkiswwlks2lem3 29891 wlkiswwlksupgr2 29897 clwlkclwwlklem2a 30017 clwlkclwwlklem2 30019 1wlkdlem1 30156 wlk2v2e 30176 eucrctshift 30262 konigsbergssiedgw 30269 wrdfd 32918 wrdres 32919 pfxf1 32926 s3f1 32931 ccatf1 32933 swrdrn3 32940 cycpmcl 33136 tocyc01 33138 cycpmco2rn 33145 cycpmrn 33163 tocyccntz 33164 cycpmconjslem2 33175 unitprodclb 33417 sseqf 34394 fiblem 34400 ofcccat 34558 signstcl 34580 signstf 34581 signstfvn 34584 signsvtn0 34585 signstres 34590 signsvtp 34598 signsvtn 34599 signsvfpn 34600 signsvfnn 34601 signshf 34603 revwlk 35130 mvrsfpw 35511 frlmfzowrdb 42514 amgm2d 44211 amgm3d 44212 amgm4d 44213 lswn0 47431 amgmw2d 49323 |
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