Theorem lcfls1c 42513

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 1105	. 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 42511	. 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 42468	. . 3 ⊢ (𝐺 ∈ 𝐷 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
6	5	anbi1i 636	. 2 ⊢ ((𝐺 ∈ 𝐷 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ ((𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
7	1, 3, 6	3bitr4i 306	1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐷 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412   ⊆ wss 3898  ‘cfv 6527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535

This theorem is used by:  lclkrslem1  42514  lclkrslem2  42515