Theorem 2wlklem 16629

Description: Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Assertion

Ref	Expression
2wlklem	|- ( A. k e. { 0 , 1 } ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )

Distinct variable groups:   k, E   k, F   P, k

Proof of Theorem 2wlklem

Step	Hyp	Ref	Expression
1		c0ex 8320	. 2 |- 0 e. _V
2		1ex 8321	. 2 |- 1 e. _V
3		2fveq3 5700	. . 3 |- ( k = 0 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 0 ) ) )
4		fveq2 5695	. . . 4 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
5		fv0p1e1 9420	. . . 4 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
6	4, 5	preq12d 3796	. . 3 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
7	3, 6	eqeq12d 2253	. 2 |- ( k = 0 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
8		2fveq3 5700	. . 3 |- ( k = 1 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 1 ) ) )
9		fveq2 5695	. . . 4 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
10		oveq1 6092	. . . . . 6 |- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
11		1p1e2 9422	. . . . . 6 |- ( 1 + 1 ) = 2
12	10, 11	eqtrdi 2287	. . . . 5 |- ( k = 1 -> ( k + 1 ) = 2 )
13	12	fveq2d 5699	. . . 4 |- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
14	9, 13	preq12d 3796	. . 3 |- ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
15	8, 14	eqeq12d 2253	. 2 |- ( k = 1 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
16	1, 2, 7, 15	ralpr 3764	1 |- ( A. k e. { 0 , 1 } ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   A.wral 2528   {cpr 3710   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   2c2 9356

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-mulcl 8277  ax-addcom 8279  ax-i2m1 8284  ax-0id 8287

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-2 9364

This theorem is used by:  upgr2wlkdc  16630