Theorem revfv 12868

Description: Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)

Assertion

Ref	Expression
revfv	|- ( ( W e. Word A /\ X e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )

Proof of Theorem revfv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		revval 12865	. . 3 |- ( W e. Word A -> (reverse ` W ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
2	1	fveq1d 5867	. 2 |- ( W e. Word A -> ( (reverse ` W ) ` X ) = ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) )
3		oveq2 6298	. . . 4 |- ( x = X -> ( ( ( # ` W ) - 1 ) - x ) = ( ( ( # ` W ) - 1 ) - X ) )
4	3	fveq2d 5869	. . 3 |- ( x = X -> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
5		eqid 2451	. . 3 |- ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) )
6		fvex 5875	. . 3 |- ( W ` ( ( ( # ` W ) - 1 ) - X ) ) e. _V
7	4, 5, 6	fvmpt 5948	. 2 |- ( X e. ( 0..^ ( # ` W ) ) -> ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
8	2, 7	sylan9eq 2505	1 |- ( ( W e. Word A /\ X e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540   - cmin 9860  ..^cfzo 11915   #chash 12515  Word cword 12656  reversecreverse 12662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-reverse 12670

This theorem is referenced by:  revs1  12870  revccat  12871  revrev  12872  revco  12931