Theorem elqtop3

Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015)

	Ref	Expression
Assertion	elqtop3	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝐴 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝐴 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) ) )

Proof

Step	Hyp	Ref	Expression
1		toponuni	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
2		eqimss	⊢ ( 𝑋 = ∪ 𝐽 → 𝑋 ⊆ ∪ 𝐽 )
3	1 2	syl	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 ⊆ ∪ 𝐽 )
4	3	adantr	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → 𝑋 ⊆ ∪ 𝐽 )
5		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
6	5	elqtop	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ∧ 𝑋 ⊆ ∪ 𝐽 ) → ( 𝐴 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝐴 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) ) )
7	4 6	mpd3an3	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝐴 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝐴 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) ) )