Theorem 2wlklem 16230

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 8173	. 2 |- 0 e. _V
2		1ex 8174	. 2 |- 1 e. _V
3		2fveq3 5644	. . 3 |- ( k = 0 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 0 ) ) )
4		fveq2 5639	. . . 4 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
5		fv0p1e1 9258	. . . 4 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
6	4, 5	preq12d 3756	. . 3 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
7	3, 6	eqeq12d 2246	. 2 |- ( k = 0 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
8		2fveq3 5644	. . 3 |- ( k = 1 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 1 ) ) )
9		fveq2 5639	. . . 4 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
10		oveq1 6025	. . . . . 6 |- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
11		1p1e2 9260	. . . . . 6 |- ( 1 + 1 ) = 2
12	10, 11	eqtrdi 2280	. . . . 5 |- ( k = 1 -> ( k + 1 ) = 2 )
13	12	fveq2d 5643	. . . 4 |- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
14	9, 13	preq12d 3756	. . 3 |- ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
15	8, 14	eqeq12d 2246	. 2 |- ( k = 1 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
16	1, 2, 7, 15	ralpr 3724	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 ) } ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   A.wral 2510   {cpr 3670   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   2c2 9194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-mulcl 8130  ax-addcom 8132  ax-i2m1 8137  ax-0id 8140

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021  df-2 9202

This theorem is referenced by:  upgr2wlkdc  16231