Theorem dicopelval2

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 20-Feb-2014)

	Ref	Expression
Hypotheses	dicval.l	⊢ ≤ = ( le ‘ 𝐾 )
	dicval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dicval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dicval.p	⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
	dicval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dicval.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dicval.i	⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
	dicval2.g	⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
	dicelval2.f	⊢ 𝐹 ∈ V
	dicelval2.s	⊢ 𝑆 ∈ V
Assertion	dicopelval2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dicval.l	⊢ ≤ = ( le ‘ 𝐾 )
2		dicval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		dicval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dicval.p	⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
5		dicval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		dicval.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
7		dicval.i	⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
8		dicval2.g	⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
9		dicelval2.f	⊢ 𝐹 ∈ V
10		dicelval2.s	⊢ 𝑆 ∈ V
11	1 2 3 4 5 6 7 9 10	dicopelval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) )
12	8	fveq2i	⊢ ( 𝑆 ‘ 𝐺 ) = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) )
13	12	eqeq2i	⊢ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ↔ 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) )
14	13	anbi1i	⊢ ( ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) )
15	11 14	bitr4di	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) )