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Metamath Proof Explorer

Theorem diafn

Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013)

Ref Expression
Hypotheses diafn.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
diafn.l ⊢ ≤ = ( le ‘ 𝐾 )
diafn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
diafn.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion diafn ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } )

Proof

Step Hyp Ref Expression
1 diafn.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 diafn.l ⊢ ≤ = ( le ‘ 𝐾 )
3 diafn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 diafn.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
5 fvex ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∈ V
6 5 rabex ⊢ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑦 } ∈ V
7 eqid ⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑦 } ) = ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑦 } )
8 6 7 fnmpti ⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑦 } ) Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 }
9 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
10 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
11 1 2 3 9 10 4 diafval ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑦 } ) )
12 11 fneq1d ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ↔ ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ≤ 𝑦 } ) Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ) )
13 8 12 mpbiri ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } )