Theorem lsw 13351

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 e. 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 3212	. 2 |- ( W e. X -> W e. _V )
2		fvex 6201	. 2 |- ( W ` ( ( # ` W ) - 1 ) ) e. _V
3		id 22	. . . 4 |- ( w = W -> w = W )
4		fveq2 6191	. . . . 5 |- ( w = W -> ( # ` w ) = ( # ` W ) )
5	4	oveq1d 6665	. . . 4 |- ( w = W -> ( ( # ` w ) - 1 ) = ( ( # ` W ) - 1 ) )
6	3, 5	fveq12d 6197	. . 3 |- ( w = W -> ( w ` ( ( # ` w ) - 1 ) ) = ( W ` ( ( # ` W ) - 1 ) ) )
7		df-lsw 13300	. . 3 |- lastS = ( w e. _V |-> ( w ` ( ( # ` w ) - 1 ) ) )
8	6, 7	fvmptg 6280	. 2 |- ( ( W e. _V /\ ( W ` ( ( # ` W ) - 1 ) ) e. _V ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
9	1, 2, 8	sylancl 694	1 |- ( W e. X -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   1c1 9937   - cmin 10266   #chash 13117   lastS clsw 13292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-lsw 13300

This theorem is referenced by:  lsw0  13352  lsw1  13354  lswcl  13355  ccatval1lsw  13368  lswccatn0lsw  13373  swrd0fvlsw  13443  swrdlsw  13452  swrdccatwrd  13468  repswlsw  13529  lswcshw  13561  lswco  13584  lsws2  13649  lsws3  13650  lsws4  13651  wrdl2exs2  13690  swrd2lsw  13695  psgnunilem5  17914  wlkonwlk1l  26559  wwlksnext  26788  wwlksnredwwlkn  26790  wwlksnextproplem2  26805  clwlkclwwlklem2a1  26893  clwlkclwwlklem2a3  26895  clwlkclwwlklem2a4  26898  clwlkclwwlklem2  26901  clwwlksn2  26910  clwwlksel  26914  clwwlksf  26915  clwwisshclwwslem  26927  numclwwlkovf2exlem2  27212  numclwlk1lem2f1  27227  numclwlk1lem2fo  27228  iwrdsplit  30449  signsvtn0  30647  signstfveq0  30654  lswn0  41380  pfxfvlsw  41403