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Theorem lsw 14514
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 (π‘Š ∈ 𝑋 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))

Proof of Theorem lsw
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
2 fvex 6905 . 2 (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) ∈ V
3 id 22 . . . 4 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
4 fveq2 6892 . . . . 5 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
54oveq1d 7424 . . . 4 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) βˆ’ 1) = ((β™―β€˜π‘Š) βˆ’ 1))
63, 5fveq12d 6899 . . 3 (𝑀 = π‘Š β†’ (π‘€β€˜((β™―β€˜π‘€) βˆ’ 1)) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
7 df-lsw 14513 . . 3 lastS = (𝑀 ∈ V ↦ (π‘€β€˜((β™―β€˜π‘€) βˆ’ 1)))
86, 7fvmptg 6997 . 2 ((π‘Š ∈ V ∧ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) ∈ V) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
91, 2, 8sylancl 587 1 (π‘Š ∈ 𝑋 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  β€˜cfv 6544  (class class class)co 7409  1c1 11111   βˆ’ cmin 11444  β™―chash 14290  lastSclsw 14512
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 5300  ax-nul 5307  ax-pr 5428
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-lsw 14513
This theorem is referenced by:  lsw0  14515  lsw1  14517  lswcl  14518  ccatval1lsw  14534  lswccatn0lsw  14541  swrdlsw  14617  pfxfvlsw  14645  repswlsw  14732  lswcshw  14765  lswco  14790  lsws2  14855  lsws3  14856  lsws4  14857  wrdl2exs2  14897  swrd2lsw  14903  psgnunilem5  19362  wlkonwlk1l  28920  wwlknlsw  29101  wwlksnext  29147  wwlksnredwwlkn  29149  wwlksnextproplem2  29164  clwlkclwwlklem2a1  29245  clwlkclwwlklem2a3  29247  clwlkclwwlklem2a4  29250  clwlkclwwlklem2  29253  clwwisshclwwslem  29267  clwwlknlbonbgr1  29292  clwwlkn2  29297  clwwlkel  29299  clwwlkf  29300  clwwlkwwlksb  29307  clwwlknonex2lem2  29361  2clwwlk2clwwlklem  29599  numclwwlk1lem2f1  29610  pfxlsw2ccat  32116  iwrdsplit  33386  signsvtn0  33581  signstfveq0  33588  lswn0  46112
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