Theorem lsw 14195

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	⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))

Proof of Theorem lsw

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		fvex 6769	. 2 ⊢ (𝑊‘((♯‘𝑊) − 1)) ∈ V
3		id 22	. . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
4		fveq2 6756	. . . . 5 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
5	4	oveq1d 7270	. . . 4 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) − 1) = ((♯‘𝑊) − 1))
6	3, 5	fveq12d 6763	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤‘((♯‘𝑤) − 1)) = (𝑊‘((♯‘𝑊) − 1)))
7		df-lsw 14194	. . 3 ⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
8	6, 7	fvmptg 6855	. 2 ⊢ ((𝑊 ∈ V ∧ (𝑊‘((♯‘𝑊) − 1)) ∈ V) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
9	1, 2, 8	sylancl 585	1 ⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ‘cfv 6418  (class class class)co 7255  1c1 10803   − cmin 11135  ♯chash 13972  lastSclsw 14193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-lsw 14194

This theorem is referenced by:  lsw0  14196  lsw1  14198  lswcl  14199  ccatval1lsw  14217  lswccatn0lsw  14224  swrdlsw  14308  pfxfvlsw  14336  repswlsw  14423  lswcshw  14456  lswco  14480  lsws2  14545  lsws3  14546  lsws4  14547  wrdl2exs2  14587  swrd2lsw  14593  psgnunilem5  19017  wlkonwlk1l  27933  wwlknlsw  28113  wwlksnext  28159  wwlksnredwwlkn  28161  wwlksnextproplem2  28176  clwlkclwwlklem2a1  28257  clwlkclwwlklem2a3  28259  clwlkclwwlklem2a4  28262  clwlkclwwlklem2  28265  clwwisshclwwslem  28279  clwwlknlbonbgr1  28304  clwwlkn2  28309  clwwlkel  28311  clwwlkf  28312  clwwlkwwlksb  28319  clwwlknonex2lem2  28373  2clwwlk2clwwlklem  28611  numclwwlk1lem2f1  28622  pfxlsw2ccat  31126  iwrdsplit  32254  signsvtn0  32449  signstfveq0  32456  lswn0  44784