Theorem ptuniconst

Description: The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015)

	Ref	Expression
Hypotheses	ptuniconst.2	⊢ 𝐽 = ( ∏t ‘ ( 𝐴 × { 𝑅 } ) )
	ptuniconst.1	⊢ 𝑋 = ∪ 𝑅
Assertion	ptuniconst	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top ) → ( 𝑋 ↑m 𝐴 ) = ∪ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		ptuniconst.2	⊢ 𝐽 = ( ∏t ‘ ( 𝐴 × { 𝑅 } ) )
2		ptuniconst.1	⊢ 𝑋 = ∪ 𝑅
3	2	toptopon	⊢ ( 𝑅 ∈ Top ↔ 𝑅 ∈ ( TopOn ‘ 𝑋 ) )
4	1	pttoponconst	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ ( TopOn ‘ 𝑋 ) ) → 𝐽 ∈ ( TopOn ‘ ( 𝑋 ↑m 𝐴 ) ) )
5	3 4	sylan2b	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top ) → 𝐽 ∈ ( TopOn ‘ ( 𝑋 ↑m 𝐴 ) ) )
6		toponuni	⊢ ( 𝐽 ∈ ( TopOn ‘ ( 𝑋 ↑m 𝐴 ) ) → ( 𝑋 ↑m 𝐴 ) = ∪ 𝐽 )
7	5 6	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top ) → ( 𝑋 ↑m 𝐴 ) = ∪ 𝐽 )