Theorem revfv 12403

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 12400	. . 3 |- ( W e. Word A -> (reverse ` W ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
2	1	fveq1d 5693	. 2 |- ( W e. Word A -> ( (reverse ` W ) ` X ) = ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) )
3		oveq2 6099	. . . 4 |- ( x = X -> ( ( ( # ` W ) - 1 ) - x ) = ( ( ( # ` W ) - 1 ) - X ) )
4	3	fveq2d 5695	. . 3 |- ( x = X -> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
5		eqid 2443	. . 3 |- ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) )
6		fvex 5701	. . 3 |- ( W ` ( ( ( # ` W ) - 1 ) - X ) ) e. _V
7	4, 5, 6	fvmpt 5774	. 2 |- ( X e. ( 0..^ ( # ` W ) ) -> ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
8	2, 7	sylan9eq 2495	1 |- ( ( W e. Word A /\ X e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   e. cmpt 4350   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283   - cmin 9595  ..^cfzo 11548   #chash 12103  Word cword 12221  reversecreverse 12227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-reverse 12235

This theorem is referenced by:  revs1  12405  revccat  12406  revrev  12407  revco  12462