Theorem revfv 12694

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 12691	. . 3 |- ( W e. Word A -> (reverse ` W ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) )
2	1	fveq1d 5866	. 2 |- ( W e. Word A -> ( (reverse ` W ) ` X ) = ( ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) ` X ) )
3		oveq2 6290	. . . 4 |- ( x = X -> ( ( ( # ` W ) - 1 ) - x ) = ( ( ( # ` W ) - 1 ) - X ) )
4	3	fveq2d 5868	. . 3 |- ( x = X -> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )
5		eqid 2467	. . 3 |- ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) ) = ( x e. ( 0..^ ( # ` W ) ) |-> ( W ` ( ( ( # ` W ) - 1 ) - x ) ) )
6		fvex 5874	. . 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 2528	1 |- ( ( W e. Word A /\ X e. ( 0..^ ( # ` W ) ) ) -> ( (reverse ` W ) ` X ) = ( W ` ( ( ( # ` W ) - 1 ) - X ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489   - cmin 9801  ..^cfzo 11788   #chash 12367  Word cword 12494  reversecreverse 12500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-reverse 12508

This theorem is referenced by:  revs1  12696  revccat  12697  revrev  12698  revco  12757