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 ) ) )