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Metamath Proof Explorer

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 ) ) )