Theorem 2wlklem 16095

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 8148	. 2 |- 0 e. _V
2		1ex 8149	. 2 |- 1 e. _V
3		2fveq3 5634	. . 3 |- ( k = 0 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 0 ) ) )
4		fveq2 5629	. . . 4 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
5		fv0p1e1 9233	. . . 4 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
6	4, 5	preq12d 3751	. . 3 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
7	3, 6	eqeq12d 2244	. 2 |- ( k = 0 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
8		2fveq3 5634	. . 3 |- ( k = 1 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 1 ) ) )
9		fveq2 5629	. . . 4 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
10		oveq1 6014	. . . . . 6 |- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
11		1p1e2 9235	. . . . . 6 |- ( 1 + 1 ) = 2
12	10, 11	eqtrdi 2278	. . . . 5 |- ( k = 1 -> ( k + 1 ) = 2 )
13	12	fveq2d 5633	. . . 4 |- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
14	9, 13	preq12d 3751	. . 3 |- ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
15	8, 14	eqeq12d 2244	. 2 |- ( k = 1 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
16	1, 2, 7, 15	ralpr 3721	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 1395   A.wral 2508   {cpr 3667   ` cfv 5318  (class class class)co 6007   0cc0 8007   1c1 8008   + caddc 8010   2c2 9169

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-1cn 8100  ax-icn 8102  ax-addcl 8103  ax-mulcl 8105  ax-addcom 8107  ax-i2m1 8112  ax-0id 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010  df-2 9177

This theorem is referenced by:  upgr2wlkdc  16096