Theorem pcoptcl

Description: A constant function is a path from Y to itself. (Contributed by Mario Carneiro, 12-Feb-2015) (Revised by Mario Carneiro, 19-Mar-2015)

	Ref	Expression
Hypothesis	pcopt.1	⊢ 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } )
Assertion	pcoptcl	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		pcopt.1	⊢ 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } )
2		iitopon	⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) )
3		cnconst2	⊢ ( ( II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ∧ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( ( 0 [,] 1 ) × { 𝑌 } ) ∈ ( II Cn 𝐽 ) )
4	2 3	mp3an1	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( ( 0 [,] 1 ) × { 𝑌 } ) ∈ ( II Cn 𝐽 ) )
5	1 4	eqeltrid	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → 𝑃 ∈ ( II Cn 𝐽 ) )
6	1	fveq1i	⊢ ( 𝑃 ‘ 0 ) = ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ 0 )
7		simpr	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → 𝑌 ∈ 𝑋 )
8		0elunit	⊢ 0 ∈ ( 0 [,] 1 )
9		fvconst2g	⊢ ( ( 𝑌 ∈ 𝑋 ∧ 0 ∈ ( 0 [,] 1 ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ 0 ) = 𝑌 )
10	7 8 9	sylancl	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ 0 ) = 𝑌 )
11	6 10	syl5eq	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( 𝑃 ‘ 0 ) = 𝑌 )
12	1	fveq1i	⊢ ( 𝑃 ‘ 1 ) = ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ 1 )
13		1elunit	⊢ 1 ∈ ( 0 [,] 1 )
14		fvconst2g	⊢ ( ( 𝑌 ∈ 𝑋 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ 1 ) = 𝑌 )
15	7 13 14	sylancl	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ 1 ) = 𝑌 )
16	12 15	syl5eq	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( 𝑃 ‘ 1 ) = 𝑌 )
17	5 11 16	3jca	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑌 ∈ 𝑋 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) )