Theorem lcfls1c 40863

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

Hypotheses

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

Assertion

Ref	Expression
lcfls1c	⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐷 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))

Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ⊥ ,𝑓   𝑄,𝑓

Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem lcfls1c

Step	Hyp	Ref	Expression
1		df-3an 1086	. 2 ⊢ ((𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ ((𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
2		lcfls1.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}
3	2	lcfls1lem 40861	. 2 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
4		lcfls1c.c	. . . 4 ⊢ 𝐷 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
5	4	lcfl1lem 40818	. . 3 ⊢ (𝐺 ∈ 𝐷 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
6	5	anbi1i 623	. 2 ⊢ ((𝐺 ∈ 𝐷 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ ((𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
7	1, 3, 6	3bitr4i 303	1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐷 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3424   ⊆ wss 3940  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541

This theorem is referenced by:  lclkrslem1  40864  lclkrslem2  40865