Theorem dibelval1st

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

	Ref	Expression
Hypotheses	dibelval1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dibelval1.l	⊢ ≤ = ( le ‘ 𝐾 )
	dibelval1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dibelval1.j	⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
	dibelval1.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dibelval1st	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		dibelval1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dibelval1.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dibelval1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dibelval1.j	⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
5		dibelval1.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		eqid	⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
8	1 2 3 6 7 4 5	dibval2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
9	8	eleq2d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) )
10	9	biimp3a	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
11		xp1st	⊢ ( 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) )
12	10 11	syl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) )