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Mirrors > Home > MPE Home > Th. List > wrdfin | Structured version Visualization version GIF version |
Description: A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.) |
Ref | Expression |
---|---|
wrdfin | ⊢ (𝑊 ∈ Word 𝑆 → 𝑊 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdfn 14370 | . 2 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊 Fn (0..^(♯‘𝑊))) | |
2 | fzofi 13833 | . 2 ⊢ (0..^(♯‘𝑊)) ∈ Fin | |
3 | fnfi 9083 | . 2 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ (0..^(♯‘𝑊)) ∈ Fin) → 𝑊 ∈ Fin) | |
4 | 1, 2, 3 | sylancl 586 | 1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Fn wfn 6488 ‘cfv 6493 (class class class)co 7351 Fincfn 8841 0cc0 11009 ..^cfzo 13521 ♯chash 14184 Word cword 14356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-fzo 13522 df-hash 14185 df-word 14357 |
This theorem is referenced by: lencl 14375 lennncl 14376 lswlgt0cl 14411 ccatval2 14420 pfxtrcfv 14539 pfxlswccat 14559 wrdeqs1cat 14566 wrdind 14568 wrd2ind 14569 ofccat 14814 gsumwmhm 18615 psgnunilem5 19235 psgnunilem4 19238 efgsres 19479 efgredlem 19488 wlkp1lem2 28451 upgrwlkdvdelem 28513 eupth2eucrct 28990 cycpmconjslem2 31830 |
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