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Metamath Proof Explorer

Theorem eltg2

Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006) (Revised by Mario Carneiro, 10-Jan-2015)

Ref Expression
Assertion eltg2 ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )

Proof

Step Hyp Ref Expression
1 tgval2 ⊢ ( 𝐵 ∈ 𝑉 → ( topGen ‘ 𝐵 ) = { 𝑧 ∣ ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) } )
2 1 eleq2d ⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ 𝐴 ∈ { 𝑧 ∣ ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) } ) )
3 elex ⊢ ( 𝐴 ∈ { 𝑧 ∣ ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) } → 𝐴 ∈ V )
4 3 adantl ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ { 𝑧 ∣ ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) } ) → 𝐴 ∈ V )
5 uniexg ⊢ ( 𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V )
6 ssexg ⊢ ( ( 𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ V ) → 𝐴 ∈ V )
7 5 6 sylan2 ⊢ ( ( 𝐴 ⊆ ∪ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ V )
8 7 ancoms ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝐵 ) → 𝐴 ∈ V )
9 8 adantrr ⊢ ( ( 𝐵 ∈ 𝑉 ∧ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) → 𝐴 ∈ V )
10 sseq1 ⊢ ( 𝑧 = 𝐴 → ( 𝑧 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ 𝐵 ) )
11 sseq2 ⊢ ( 𝑧 = 𝐴 → ( 𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝐴 ) )
12 11 anbi2d ⊢ ( 𝑧 = 𝐴 → ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ↔ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) )
13 12 rexbidv ⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ↔ ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) )
14 13 raleqbi1dv ⊢ ( 𝑧 = 𝐴 → ( ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) )
15 10 14 anbi12d ⊢ ( 𝑧 = 𝐴 → ( ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
16 15 elabg ⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝑧 ∣ ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) } ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
17 4 9 16 pm5.21nd ⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ { 𝑧 ∣ ( 𝑧 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝑧 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧 ) ) } ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
18 2 17 bitrd ⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )