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Theorem lsw 14511
Description: Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
Assertion
Ref Expression
lsw (𝑊𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))

Proof of Theorem lsw
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝑊𝑋𝑊 ∈ V)
2 fvex 6902 . 2 (𝑊‘((♯‘𝑊) − 1)) ∈ V
3 id 22 . . . 4 (𝑤 = 𝑊𝑤 = 𝑊)
4 fveq2 6889 . . . . 5 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
54oveq1d 7421 . . . 4 (𝑤 = 𝑊 → ((♯‘𝑤) − 1) = ((♯‘𝑊) − 1))
63, 5fveq12d 6896 . . 3 (𝑤 = 𝑊 → (𝑤‘((♯‘𝑤) − 1)) = (𝑊‘((♯‘𝑊) − 1)))
7 df-lsw 14510 . . 3 lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
86, 7fvmptg 6994 . 2 ((𝑊 ∈ V ∧ (𝑊‘((♯‘𝑊) − 1)) ∈ V) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
91, 2, 8sylancl 587 1 (𝑊𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cfv 6541  (class class class)co 7406  1c1 11108  cmin 11441  chash 14287  lastSclsw 14509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6493  df-fun 6543  df-fv 6549  df-ov 7409  df-lsw 14510
This theorem is referenced by:  lsw0  14512  lsw1  14514  lswcl  14515  ccatval1lsw  14531  lswccatn0lsw  14538  swrdlsw  14614  pfxfvlsw  14642  repswlsw  14729  lswcshw  14762  lswco  14787  lsws2  14852  lsws3  14853  lsws4  14854  wrdl2exs2  14894  swrd2lsw  14900  psgnunilem5  19357  wlkonwlk1l  28910  wwlknlsw  29091  wwlksnext  29137  wwlksnredwwlkn  29139  wwlksnextproplem2  29154  clwlkclwwlklem2a1  29235  clwlkclwwlklem2a3  29237  clwlkclwwlklem2a4  29240  clwlkclwwlklem2  29243  clwwisshclwwslem  29257  clwwlknlbonbgr1  29282  clwwlkn2  29287  clwwlkel  29289  clwwlkf  29290  clwwlkwwlksb  29297  clwwlknonex2lem2  29351  2clwwlk2clwwlklem  29589  numclwwlk1lem2f1  29600  pfxlsw2ccat  32104  iwrdsplit  33375  signsvtn0  33570  signstfveq0  33577  lswn0  46099
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