Metamath Proof Explorer


Theorem isfne4

Description: The predicate " B is finer than A " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015)

Ref Expression
Hypotheses isfne.1 𝑋 = 𝐴
isfne.2 𝑌 = 𝐵
Assertion isfne4 ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 isfne.1 𝑋 = 𝐴
2 isfne.2 𝑌 = 𝐵
3 fnerel Rel Fne
4 3 brrelex2i ( 𝐴 Fne 𝐵𝐵 ∈ V )
5 simpl ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝑋 = 𝑌 )
6 5 1 2 3eqtr3g ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐴 = 𝐵 )
7 fvex ( topGen ‘ 𝐵 ) ∈ V
8 7 ssex ( 𝐴 ⊆ ( topGen ‘ 𝐵 ) → 𝐴 ∈ V )
9 8 adantl ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐴 ∈ V )
10 9 uniexd ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐴 ∈ V )
11 6 10 eqeltrrd ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐵 ∈ V )
12 uniexb ( 𝐵 ∈ V ↔ 𝐵 ∈ V )
13 11 12 sylibr ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) → 𝐵 ∈ V )
14 1 2 isfne ( 𝐵 ∈ V → ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌 ∧ ∀ 𝑥𝐴 𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) ) ) )
15 dfss3 ( 𝐴 ⊆ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥𝐴 𝑥 ∈ ( topGen ‘ 𝐵 ) )
16 eltg ( 𝐵 ∈ V → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ 𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) ) )
17 16 ralbidv ( 𝐵 ∈ V → ( ∀ 𝑥𝐴 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥𝐴 𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) ) )
18 15 17 syl5bb ( 𝐵 ∈ V → ( 𝐴 ⊆ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥𝐴 𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) ) )
19 18 anbi2d ( 𝐵 ∈ V → ( ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) ↔ ( 𝑋 = 𝑌 ∧ ∀ 𝑥𝐴 𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) ) ) )
20 14 19 bitr4d ( 𝐵 ∈ V → ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) ) )
21 4 13 20 pm5.21nii ( 𝐴 Fne 𝐵 ↔ ( 𝑋 = 𝑌𝐴 ⊆ ( topGen ‘ 𝐵 ) ) )