Theorem 2wlkdlem8

Description: Lemma 8 for 2wlkd . (Contributed by AV, 14-Feb-2021)

	Ref	Expression
Hypotheses	2wlkd.p	|- P = <" A B C ">
	2wlkd.f	|- F = <" J K ">
	2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
	2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
	2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
Assertion	2wlkdlem8	|- ( ph -> ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	|- P = <" A B C ">
2		2wlkd.f	|- F = <" J K ">
3		2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4		2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
5		2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6	1 2 3 4 5	2wlkdlem7	|- ( ph -> ( J e. _V /\ K e. _V ) )
7		s2fv0	|- ( J e. _V -> ( <" J K "> ` 0 ) = J )
8		s2fv1	|- ( K e. _V -> ( <" J K "> ` 1 ) = K )
9	7 8	anim12i	|- ( ( J e. _V /\ K e. _V ) -> ( ( <" J K "> ` 0 ) = J /\ ( <" J K "> ` 1 ) = K ) )
10	6 9	syl	|- ( ph -> ( ( <" J K "> ` 0 ) = J /\ ( <" J K "> ` 1 ) = K ) )
11	2	fveq1i	|- ( F ` 0 ) = ( <" J K "> ` 0 )
12	11	eqeq1i	|- ( ( F ` 0 ) = J <-> ( <" J K "> ` 0 ) = J )
13	2	fveq1i	|- ( F ` 1 ) = ( <" J K "> ` 1 )
14	13	eqeq1i	|- ( ( F ` 1 ) = K <-> ( <" J K "> ` 1 ) = K )
15	12 14	anbi12i	|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) <-> ( ( <" J K "> ` 0 ) = J /\ ( <" J K "> ` 1 ) = K ) )
16	10 15	sylibr	|- ( ph -> ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) )