Theorem lsw 13204

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 3185	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		fvex 6113	. 2 ⊢ (𝑊‘((#‘𝑊) − 1)) ∈ V
3		id 22	. . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
4		fveq2 6103	. . . . 5 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
5	4	oveq1d 6564	. . . 4 ⊢ (𝑤 = 𝑊 → ((#‘𝑤) − 1) = ((#‘𝑊) − 1))
6	3, 5	fveq12d 6109	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤‘((#‘𝑤) − 1)) = (𝑊‘((#‘𝑊) − 1)))
7		df-lsw 13155	. . 3 ⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((#‘𝑤) − 1)))
8	6, 7	fvmptg 6189	. 2 ⊢ ((𝑊 ∈ V ∧ (𝑊‘((#‘𝑊) − 1)) ∈ V) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
9	1, 2, 8	sylancl 693	1 ⊢ (𝑊 ∈ 𝑋 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  1c1 9816   − cmin 10145  #chash 12979   lastS clsw 13147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-lsw 13155

This theorem is referenced by:  lsw0  13205  lsw1  13207  lswcl  13208  ccatval1lsw  13221  lswccatn0lsw  13226  swrd0fvlsw  13295  swrdlsw  13304  swrdccatwrd  13320  repswlsw  13380  lswcshw  13412  lswco  13435  lsws2  13499  lsws3  13500  lsws4  13501  wrdl2exs2  13538  swrd2lsw  13543  psgnunilem5  17737  wwlknext  26252  wwlknredwwlkn  26254  wwlkextproplem2  26270  clwwlkgt0  26299  clwwlkn2  26303  clwlkisclwwlklem2a1  26307  clwlkisclwwlklem2a3  26309  clwlkisclwwlklem2a4  26312  clwlkisclwwlklem1  26315  clwwlkel  26321  clwwlkf  26322  clwwisshclwwlem  26334  numclwwlkovf2ex  26613  numclwlk1lem2f1  26621  numclwlk1lem2fo  26622  iwrdsplit  29776  signsvtn0  29973  signstfveq0  29980  lswn0  40242  pfxfvlsw  40266  wlkOnwlk1l  40871  wwlksnext  41099  wwlksnredwwlkn  41101  wwlksnextproplem2  41116  clwlkclwwlklem2a1  41201  clwlkclwwlklem2a3  41203  clwlkclwwlklem2a4  41206  clwlkclwwlklem2  41209  clwwlksn2  41217  clwwlksel  41221  clwwlksf  41222  clwwisshclwwslem  41234  av-numclwwlkovf2ex  41517  av-numclwlk1lem2f1  41524  av-numclwlk1lem2fo  41525