Theorem phtpyhtpy

Description: A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	isphtpy.2	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
	isphtpy.3	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
Assertion	phtpyhtpy	⊢ ( 𝜑 → ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ⊆ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		isphtpy.2	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
2		isphtpy.3	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
3	1 2	isphtpy	⊢ ( 𝜑 → ( ℎ ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ↔ ( ℎ ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) ∧ ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 ℎ 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 ℎ 𝑠 ) = ( 𝐹 ‘ 1 ) ) ) ) )
4		simpl	⊢ ( ( ℎ ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) ∧ ∀ 𝑠 ∈ ( 0 [,] 1 ) ( ( 0 ℎ 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 ℎ 𝑠 ) = ( 𝐹 ‘ 1 ) ) ) → ℎ ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )
5	3 4	biimtrdi	⊢ ( 𝜑 → ( ℎ ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) → ℎ ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) ) )
6	5	ssrdv	⊢ ( 𝜑 → ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐺 ) ⊆ ( 𝐹 ( II Htpy 𝐽 ) 𝐺 ) )