Theorem diaval

Description: The partial isomorphism A for a lattice K . Definition of isomorphism map in Crawley p. 120 line 24. (Contributed by NM, 15-Oct-2013)

	Ref	Expression
Hypotheses	diaval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	diaval.l	⊢ ≤ = ( le ‘ 𝐾 )
	diaval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	diaval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	diaval.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	diaval.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	diaval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )

Proof

Step	Hyp	Ref	Expression
1		diaval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		diaval.l	⊢ ≤ = ( le ‘ 𝐾 )
3		diaval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		diaval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		diaval.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6		diaval.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 4 5 6	diafval	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )
8	7	adantr	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )
9	8	fveq1d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ‘ 𝑋 ) )
10		breq1	⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊 ) )
11	10	elrab	⊢ ( 𝑋 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
12	11	bilanri	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → 𝑋 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } )
13		breq2	⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 ↔ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) )
14	13	rabbidv	⊢ ( 𝑥 = 𝑋 → { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
15		eqid	⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } )
16	4	fvexi	⊢ 𝑇 ∈ V
17	16	rabex	⊢ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ∈ V
18	14 15 17	fvmpt	⊢ ( 𝑋 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } → ( ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
19	12 18	syl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
20	9 19	eqtrd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )