Theorem wlkiswwlks2lem2 26756

Description: Lemma 2 for wlkiswwlks2 26761. (Contributed by Alexander van der Vekens, 20-Jul-2018.)

Hypothesis

Ref	Expression
wlkiswwlks2lem.f	|- F = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) )

Assertion

Ref	Expression
wlkiswwlks2lem2	|- ( ( ( # ` P ) e. NN0 /\ I e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( F ` I ) = ( `' E ` { ( P ` I ) , ( P ` ( I + 1 ) ) } ) )

Distinct variable groups:   x, P   x, E   x, I

Allowed substitution hint:   F( x)

Proof of Theorem wlkiswwlks2lem2

Step	Hyp	Ref	Expression
1		wlkiswwlks2lem.f	. . 3 |- F = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) )
2	1	a1i 11	. 2 |- ( ( ( # ` P ) e. NN0 /\ I e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> F = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) )
3		fveq2 6191	. . . . 5 |- ( x = I -> ( P ` x ) = ( P ` I ) )
4		oveq1 6657	. . . . . 6 |- ( x = I -> ( x + 1 ) = ( I + 1 ) )
5	4	fveq2d 6195	. . . . 5 |- ( x = I -> ( P ` ( x + 1 ) ) = ( P ` ( I + 1 ) ) )
6	3, 5	preq12d 4276	. . . 4 |- ( x = I -> { ( P ` x ) , ( P ` ( x + 1 ) ) } = { ( P ` I ) , ( P ` ( I + 1 ) ) } )
7	6	fveq2d 6195	. . 3 |- ( x = I -> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) = ( `' E ` { ( P ` I ) , ( P ` ( I + 1 ) ) } ) )
8	7	adantl 482	. 2 |- ( ( ( ( # ` P ) e. NN0 /\ I e. ( 0..^ ( ( # ` P ) - 1 ) ) ) /\ x = I ) -> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) = ( `' E ` { ( P ` I ) , ( P ` ( I + 1 ) ) } ) )
9		simpr 477	. 2 |- ( ( ( # ` P ) e. NN0 /\ I e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> I e. ( 0..^ ( ( # ` P ) - 1 ) ) )
10		fvexd 6203	. 2 |- ( ( ( # ` P ) e. NN0 /\ I e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( `' E ` { ( P ` I ) , ( P ` ( I + 1 ) ) } ) e. _V )
11	2, 8, 9, 10	fvmptd 6288	1 |- ( ( ( # ` P ) e. NN0 /\ I e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( F ` I ) = ( `' E ` { ( P ` I ) , ( P ` ( I + 1 ) ) } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179   |-> cmpt 4729   `'ccnv 5113   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292  ..^cfzo 12465   #chash 13117

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-csb 3534  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

This theorem is referenced by:  wlkiswwlks2lem4  26758