wrdf - Metamath Proof Explorer

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Theorem wrdf 13858

Description: A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)

**Assertion**
Ref	Expression
wrdf	⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Proof of Theorem wrdf Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iswrd 13855	. 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆)
2		simpr 487	. . . 4 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^𝑙)⟶𝑆)
3		fnfzo0hash 13800	. . . . . 6 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → (♯‘𝑊) = 𝑙)
4	3	oveq2d 7164	. . . . 5 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → (0..^(♯‘𝑊)) = (0..^𝑙))
5	4	feq2d 6493	. . . 4 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → (𝑊:(0..^(♯‘𝑊))⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆))
6	2, 5	mpbird 259	. . 3 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^(♯‘𝑊))⟶𝑆)
7	6	rexlimiva 3279	. 2 ⊢ (∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)
8	1, 7	sylbi 219	1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 ∃wrex 3137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 0cc0 10529 ℕ₀cn0 11889 ..^cfzo 13025 ♯chash 13682 Word cword 13853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-hash 13683 df-word 13854

This theorem is referenced by: iswrdb 13859 wrddm 13860 wrdsymbcl 13867 wrdfn 13868 wrdvOLD 13870 wrdffz 13877 0wrd0 13882 wrdsymb 13885 wrdnval 13888 wrdred1 13904 wrdred1hash 13905 ccatcl 13918 ccatalpha 13939 s1dm 13954 swrdcl 13999 swrdf 14004 swrdwrdsymb 14016 pfxres 14033 cats1un 14075 revcl 14115 revlen 14116 revrev 14121 repsdf2 14132 cshwf 14154 cshinj 14165 wrdco 14185 lenco 14186 revco 14188 ccatco 14189 lswco 14193 s2dm 14244 wwlktovf 14312 ofccat 14321 gsumwsubmcl 17993 gsumsgrpccat 17996 gsumccatOLD 17997 gsumwmhm 18002 frmdss2 18020 symgtrinv 18592 psgnunilem5 18614 psgnunilem2 18615 psgnunilem3 18616 efginvrel1 18846 efgsf 18847 efgsrel 18852 efgs1b 18854 efgredlemf 18859 efgredlemd 18862 efgredlemc 18863 efgredlem 18865 frgpup3lem 18895 pgpfaclem1 19195 ablfaclem2 19200 ablfaclem3 19201 ablfac2 19203 dchrptlem1 25832 dchrptlem2 25833 trgcgrg 26293 tgcgr4 26309 wrdupgr 26862 wrdumgr 26874 vdegp1ai 27310 vdegp1bi 27311 wlkres 27444 wlkp1 27455 wlkdlem1 27456 trlf1 27472 trlreslem 27473 upgrwlkdvdelem 27509 pthdlem1 27539 pthdlem2lem 27540 uspgrn2crct 27578 wlkiswwlks2lem3 27641 wlkiswwlksupgr2 27647 clwlkclwwlklem2a 27768 clwlkclwwlklem2 27770 1wlkdlem1 27908 wlk2v2e 27928 eucrctshift 28014 konigsbergssiedgw 28021 wrdfd 30605 wrdres 30606 pfxf1 30611 s3f1 30616 ccatf1 30618 swrdrn3 30622 cycpmcl 30751 tocyc01 30753 cycpmco2rn 30760 cycpmrn 30778 tocyccntz 30779 cycpmconjslem2 30790 sseqf 31643 fiblem 31649 ofcccat 31806 signstcl 31828 signstf 31829 signstfvn 31832 signsvtn0 31833 signstres 31838 signsvtp 31846 signsvtn 31847 signsvfpn 31848 signsvfnn 31849 signshf 31851 revwlk 32364 mvrsfpw 32746 frlmfzowrdb 39133 amgm2d 40541 amgm3d 40542 amgm4d 40543 lswn0 43594 amgmw2d 44895

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