Theorem lcfls1c

Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015)

	Ref	Expression
Hypotheses	lcfls1.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) }
	lcfls1c.c	⊢ 𝐷 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
Assertion	lcfls1c	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐷 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		lcfls1.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) }
2		lcfls1c.c	⊢ 𝐷 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
3		df-3an	⊢ ( ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ↔ ( ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )
4	1	lcfls1lem	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )
5	2	lcfl1lem	⊢ ( 𝐺 ∈ 𝐷 ↔ ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) )
6	5	anbi1i	⊢ ( ( 𝐺 ∈ 𝐷 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) ↔ ( ( 𝐺 ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝐿 ‘ 𝐺 ) ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )
7	3 4 6	3bitr4i	⊢ ( 𝐺 ∈ 𝐶 ↔ ( 𝐺 ∈ 𝐷 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑄 ) )