Theorem lcfls1N

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

	Ref	Expression
Hypotheses	lcfls1.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) }
	lcfls1.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
Assertion	lcfls1N	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 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		lcfls1.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
3	1	lcfls1lem	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )
4		3anass	⊢ ( ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ↔ ( 𝐺 ∈ 𝐹 ∧ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) )
5	3 4	bitri	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) )
6	2	biantrurd	⊢ ( 𝜑 → ( ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ↔ ( 𝐺 ∈ 𝐹 ∧ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) ) )
7	5 6	bitr4id	⊢ ( 𝜑 → ( 𝐺 ∈ 𝐶 ↔ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ) )