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Metamath Proof Explorer

Theorem dibopelval2

Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses dibval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dibval2.l ⊢ ≤ = ( le ‘ 𝐾 )
dibval2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dibval2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dibval2.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
dibval2.j ⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
dibval2.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibopelval2 ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) )

Proof

Step Hyp Ref Expression
1 dibval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dibval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dibval2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dibval2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 dibval2.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6 dibval2.j ⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 dibval2.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 6 7 dibval2 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) )
9 8 eleq2d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ) )
10 opelxp ⊢ ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 ∈ { 0 } ) )
11 4 fvexi ⊢ 𝑇 ∈ V
12 11 mptex ⊢ ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) ∈ V
13 5 12 eqeltri ⊢ 0 ∈ V
14 13 elsn2 ⊢ ( 𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
15 14 anbi2i ⊢ ( ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 ∈ { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) )
16 10 15 bitri ⊢ ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( 𝐽 ‘ 𝑋 ) × { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) )
17 9 16 bitrdi ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( 𝐽 ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) )