Step |
Hyp |
Ref |
Expression |
1 |
|
istmd.1 |
⊢ 𝐹 = ( +𝑓 ‘ 𝐺 ) |
2 |
|
istmd.2 |
⊢ 𝐽 = ( TopOpen ‘ 𝐺 ) |
3 |
|
elin |
⊢ ( 𝐺 ∈ ( Mnd ∩ TopSp ) ↔ ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ) ) |
4 |
3
|
anbi1i |
⊢ ( ( 𝐺 ∈ ( Mnd ∩ TopSp ) ∧ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ↔ ( ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ) ∧ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ) |
5 |
|
fvexd |
⊢ ( 𝑓 = 𝐺 → ( TopOpen ‘ 𝑓 ) ∈ V ) |
6 |
|
simpl |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → 𝑓 = 𝐺 ) |
7 |
6
|
fveq2d |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → ( +𝑓 ‘ 𝑓 ) = ( +𝑓 ‘ 𝐺 ) ) |
8 |
7 1
|
eqtr4di |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → ( +𝑓 ‘ 𝑓 ) = 𝐹 ) |
9 |
|
id |
⊢ ( 𝑗 = ( TopOpen ‘ 𝑓 ) → 𝑗 = ( TopOpen ‘ 𝑓 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑓 = 𝐺 → ( TopOpen ‘ 𝑓 ) = ( TopOpen ‘ 𝐺 ) ) |
11 |
10 2
|
eqtr4di |
⊢ ( 𝑓 = 𝐺 → ( TopOpen ‘ 𝑓 ) = 𝐽 ) |
12 |
9 11
|
sylan9eqr |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → 𝑗 = 𝐽 ) |
13 |
12 12
|
oveq12d |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → ( 𝑗 ×t 𝑗 ) = ( 𝐽 ×t 𝐽 ) ) |
14 |
13 12
|
oveq12d |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → ( ( 𝑗 ×t 𝑗 ) Cn 𝑗 ) = ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
15 |
8 14
|
eleq12d |
⊢ ( ( 𝑓 = 𝐺 ∧ 𝑗 = ( TopOpen ‘ 𝑓 ) ) → ( ( +𝑓 ‘ 𝑓 ) ∈ ( ( 𝑗 ×t 𝑗 ) Cn 𝑗 ) ↔ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ) |
16 |
5 15
|
sbcied |
⊢ ( 𝑓 = 𝐺 → ( [ ( TopOpen ‘ 𝑓 ) / 𝑗 ] ( +𝑓 ‘ 𝑓 ) ∈ ( ( 𝑗 ×t 𝑗 ) Cn 𝑗 ) ↔ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ) |
17 |
|
df-tmd |
⊢ TopMnd = { 𝑓 ∈ ( Mnd ∩ TopSp ) ∣ [ ( TopOpen ‘ 𝑓 ) / 𝑗 ] ( +𝑓 ‘ 𝑓 ) ∈ ( ( 𝑗 ×t 𝑗 ) Cn 𝑗 ) } |
18 |
16 17
|
elrab2 |
⊢ ( 𝐺 ∈ TopMnd ↔ ( 𝐺 ∈ ( Mnd ∩ TopSp ) ∧ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ) |
19 |
|
df-3an |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ↔ ( ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ) ∧ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ) |
20 |
4 18 19
|
3bitr4i |
⊢ ( 𝐺 ∈ TopMnd ↔ ( 𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) ) |