Theorem isphtpyd

Description: Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	isphtpy.2	|- ( ph -> F e. ( II Cn J ) )
	isphtpy.3	|- ( ph -> G e. ( II Cn J ) )
	isphtpyd.1	|- ( ph -> H e. ( F ( II Htpy J ) G ) )
	isphtpyd.2	|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 H s ) = ( F ` 0 ) )
	isphtpyd.3	|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 H s ) = ( F ` 1 ) )
Assertion	isphtpyd	|- ( ph -> H e. ( F ( PHtpy ` J ) G ) )

Proof

Step	Hyp	Ref	Expression
1		isphtpy.2	|- ( ph -> F e. ( II Cn J ) )
2		isphtpy.3	|- ( ph -> G e. ( II Cn J ) )
3		isphtpyd.1	|- ( ph -> H e. ( F ( II Htpy J ) G ) )
4		isphtpyd.2	|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 H s ) = ( F ` 0 ) )
5		isphtpyd.3	|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 H s ) = ( F ` 1 ) )
6	4 5	jca	|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) )
7	6	ralrimiva	|- ( ph -> A. s e. ( 0 [,] 1 ) ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) )
8	1 2	isphtpy	|- ( ph -> ( H e. ( F ( PHtpy ` J ) G ) <-> ( H e. ( F ( II Htpy J ) G ) /\ A. s e. ( 0 [,] 1 ) ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) ) ) )
9	3 7 8	mpbir2and	|- ( ph -> H e. ( F ( PHtpy ` J ) G ) )