Metamath Proof Explorer
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 ) ) ) |