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Metamath Proof Explorer

Theorem ipotset

Description: Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015)

Ref Expression
Hypotheses ipoval.i ⊢ 𝐼 = ( toInc ‘ 𝐹 )
ipole.l ⊢ ≤ = ( le ‘ 𝐼 )
Assertion ipotset ( 𝐹 ∈ 𝑉 → ( ordTop ‘ ≤ ) = ( TopSet ‘ 𝐼 ) )

Proof

Step Hyp Ref Expression
1 ipoval.i ⊢ 𝐼 = ( toInc ‘ 𝐹 )
2 ipole.l ⊢ ≤ = ( le ‘ 𝐼 )
3 fvex ⊢ ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ∈ V
4 ipostr ⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } ) Struct ⟨ 1 , 1 1 ⟩
5 tsetid ⊢ TopSet = Slot ( TopSet ‘ ndx )
6 snsspr2 ⊢ { ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ }
7 ssun1 ⊢ { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } )
8 6 7 sstri ⊢ { ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } )
9 4 5 8 strfv ⊢ ( ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ∈ V → ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) = ( TopSet ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } ) ) )
10 3 9 ax-mp ⊢ ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) = ( TopSet ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } ) )
11 1 ipolerval ⊢ ( 𝐹 ∈ 𝑉 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } = ( le ‘ 𝐼 ) )
12 2 11 eqtr4id ⊢ ( 𝐹 ∈ 𝑉 → ≤ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } )
13 12 fveq2d ⊢ ( 𝐹 ∈ 𝑉 → ( ordTop ‘ ≤ ) = ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) )
14 eqid ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) }
15 1 14 ipoval ⊢ ( 𝐹 ∈ 𝑉 → 𝐼 = ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } ) )
16 15 fveq2d ⊢ ( 𝐹 ∈ 𝑉 → ( TopSet ‘ 𝐼 ) = ( TopSet ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐹 ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ordTop ‘ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ) ⟩ } ∪ { ⟨ ( le ‘ ndx ) , { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( { 𝑥 , 𝑦 } ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦 ) } ⟩ , ⟨ ( oc ‘ ndx ) , ( 𝑥 ∈ 𝐹 ↦ ∪ { 𝑦 ∈ 𝐹 ∣ ( 𝑦 ∩ 𝑥 ) = ∅ } ) ⟩ } ) ) )
17 10 13 16 3eqtr4a ⊢ ( 𝐹 ∈ 𝑉 → ( ordTop ‘ ≤ ) = ( TopSet ‘ 𝐼 ) )