Theorem elqtop2

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

	Ref	Expression
Hypothesis	qtoptop.1	⊢ 𝑋 = ∪ 𝐽
Assertion	elqtop2	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝐴 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝐴 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) ) )

Proof

Step	Hyp	Ref	Expression
1		qtoptop.1	⊢ 𝑋 = ∪ 𝐽
2		ssid	⊢ 𝑋 ⊆ 𝑋
3	1	elqtop	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐹 : 𝑋 –onto→ 𝑌 ∧ 𝑋 ⊆ 𝑋 ) → ( 𝐴 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝐴 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) ) )
4	2 3	mp3an3	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝐴 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝐴 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) ) )