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Mirrors > Home > MPE Home > Th. List > wrdfn | Structured version Visualization version GIF version |
Description: A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.) |
Ref | Expression |
---|---|
wrdfn | ⊢ (𝑊 ∈ Word 𝑆 → 𝑊 Fn (0..^(♯‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdf 13604 | . 2 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) | |
2 | 1 | ffnd 6292 | 1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊 Fn (0..^(♯‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 0cc0 10272 ..^cfzo 12784 ♯chash 13435 Word cword 13599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 |
This theorem is referenced by: wrdlndmOLD 13618 iswrdsymb 13619 wrdfin 13620 eqwrd 13647 ccatlid 13676 ccatrid 13677 swrd0lenOLD 13738 swrdidOLD 13745 pfxid 13793 pfxccat1 13811 revrev 13913 cshimadifsn 13980 revco 13985 cshco 13987 swrdco 13988 s3fn 14062 wrd2pr2op 14094 wrd3tpop 14099 wlkresOLD 27021 wlkp1lem2 27025 wwlksm1edg 27230 wwlksm1edgOLD 27231 sseqfv1 31050 sseqfn 31051 sseqfres 31054 sseqfv2 31055 signstres 31253 |
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