wrdf - Metamath Proof Explorer

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Theorem wrdf 13862

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 13859	. 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆)
2		simpr 488	. . . 4 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^𝑙)⟶𝑆)
3		fnfzo0hash 13804	. . . . . 6 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → (♯‘𝑊) = 𝑙)
4	3	oveq2d 7151	. . . . 5 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → (0..^(♯‘𝑊)) = (0..^𝑙))
5	4	feq2d 6473	. . . 4 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → (𝑊:(0..^(♯‘𝑊))⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆))
6	2, 5	mpbird 260	. . 3 ⊢ ((𝑙 ∈ ℕ₀ ∧ 𝑊:(0..^𝑙)⟶𝑆) → 𝑊:(0..^(♯‘𝑊))⟶𝑆)
7	6	rexlimiva 3240	. 2 ⊢ (∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)
8	1, 7	sylbi 220	1 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∃wrex 3107 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℕ₀cn0 11885 ..^cfzo 13028 ♯chash 13686 Word cword 13857

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858

This theorem is referenced by: iswrdb 13863 wrddm 13864 wrdsymbcl 13870 wrdfn 13871 wrdffz 13878 0wrd0 13883 wrdsymb 13885 wrdnval 13888 wrdred1 13903 wrdred1hash 13904 ccatcl 13917 ccatalpha 13938 s1dm 13953 swrdcl 13998 swrdf 14003 swrdwrdsymb 14015 pfxres 14032 cats1un 14074 revcl 14114 revlen 14115 revrev 14120 repsdf2 14131 cshwf 14153 cshinj 14164 wrdco 14184 lenco 14185 revco 14187 ccatco 14188 lswco 14192 s2dm 14243 wwlktovf 14311 ofccat 14320 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 19196 ablfaclem2 19201 ablfaclem3 19202 ablfac2 19204 dchrptlem1 25848 dchrptlem2 25849 trgcgrg 26309 tgcgr4 26325 wrdupgr 26878 wrdumgr 26890 vdegp1ai 27326 vdegp1bi 27327 wlkres 27460 wlkp1 27471 wlkdlem1 27472 trlf1 27488 trlreslem 27489 upgrwlkdvdelem 27525 pthdlem1 27555 pthdlem2lem 27556 uspgrn2crct 27594 wlkiswwlks2lem3 27657 wlkiswwlksupgr2 27663 clwlkclwwlklem2a 27783 clwlkclwwlklem2 27785 1wlkdlem1 27922 wlk2v2e 27942 eucrctshift 28028 konigsbergssiedgw 28035 wrdfd 30638 wrdres 30639 pfxf1 30644 s3f1 30649 ccatf1 30651 swrdrn3 30655 cycpmcl 30808 tocyc01 30810 cycpmco2rn 30817 cycpmrn 30835 tocyccntz 30836 cycpmconjslem2 30847 sseqf 31760 fiblem 31766 ofcccat 31923 signstcl 31945 signstf 31946 signstfvn 31949 signsvtn0 31950 signstres 31955 signsvtp 31963 signsvtn 31964 signsvfpn 31965 signsvfnn 31966 signshf 31968 revwlk 32484 mvrsfpw 32866 frlmfzowrdb 39438 amgm2d 40904 amgm3d 40905 amgm4d 40906 lswn0 43961 amgmw2d 45332

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