Metamath Proof Explorer

Theorem diaval

Description: The partial isomorphism A for a lattice K . Definition of isomorphism map in Crawley p. 120 line 24. (Contributed by NM, 15-Oct-2013)

Ref Expression
Hypotheses diaval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
diaval.l ⊢ ≤ = ( le ‘ 𝐾 )
diaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
diaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
diaval.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
diaval.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion diaval ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )

Proof

Step Hyp Ref Expression
1 diaval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 diaval.l ⊢ ≤ = ( le ‘ 𝐾 )
3 diaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 diaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 diaval.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 diaval.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 3 4 5 6 diafval ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )
8 7 adantr ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )
9 8 fveq1d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ‘ 𝑋 ) )
10 simpr ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
11 breq1 ⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊 ) )
12 11 elrab ⊢ ( 𝑋 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
13 10 12 sylibr ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → 𝑋 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } )
14 breq2 ⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 ↔ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) )
15 14 rabbidv ⊢ ( 𝑥 = 𝑋 → { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
16 eqid ⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } )
17 4 fvexi ⊢ 𝑇 ∈ V
18 17 rabex ⊢ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ∈ V
19 15 16 18 fvmpt ⊢ ( 𝑋 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } → ( ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
20 13 19 syl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
21 9 20 eqtrd ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )