Theorem revfv 12700

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 12697	. . 3 |- ( W e. Word A -> (reverse ` W ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
2	1	fveq1d 5807	. 2 |- ( W e. Word A -> ( (reverse ` W ) ` X ) = ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) )
3		oveq2 6242	. . . 4 |- ( x = X -> ( ( ( # ` W ) - 1 ) - x ) = ( ( ( # ` W ) - 1 ) - X ) )
4	3	fveq2d 5809	. . 3 |- ( x = X -> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
5		eqid 2402	. . 3 |- ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) )
6		fvex 5815	. . 3 |- ( W ` ( ( ( # ` W ) - 1 ) - X ) ) e. _V
7	4, 5, 6	fvmpt 5888	. 2 |- ( X e. ( 0..^ ( # ` W ) ) -> ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
8	2, 7	sylan9eq 2463	1 |- ( ( W e. Word A /\ X e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1405   e. wcel 1842   |-> cmpt 4452   ` cfv 5525  (class class class)co 6234   0cc0 9442   1c1 9443   - cmin 9761  ..^cfzo 11767   #chash 12359  Word cword 12490  reversecreverse 12496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-reverse 12504

This theorem is referenced by:  revs1  12702  revccat  12703  revrev  12704  revco  12763