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Metamath Proof Explorer

Theorem dibopelvalN

Description: Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014) (Revised by Mario Carneiro, 6-May-2015) (New usage is discouraged.)

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

Proof

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