Theorem lcfls1lem

Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015)

	Ref	Expression
Hypothesis	lcfls1.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) }
Assertion	lcfls1lem	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		lcfls1.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) }
2		fveq2	⊢ ( 𝑓 = 𝐺 → ( 𝐿 ‘ 𝑓 ) = ( 𝐿 ‘ 𝐺 ) )
3	2	fveq2d	⊢ ( 𝑓 = 𝐺 → ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) )
4	3	fveq2d	⊢ ( 𝑓 = 𝐺 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) )
5	4 2	eqeq12d	⊢ ( 𝑓 = 𝐺 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) )
6	3	sseq1d	⊢ ( 𝑓 = 𝐺 → ( ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ↔ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )
7	5 6	anbi12d	⊢ ( 𝑓 = 𝐺 → ( ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) )
8	7 1	elrab2	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) )
9		3anass	⊢ ( ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ↔ ( 𝐺 ∈ 𝐹 ∧ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) )
10	8 9	bitr4i	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )