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Theorem lcfls1c 38825
Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
Hypotheses
Ref Expression
lcfls1.c 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
lcfls1c.c 𝐷 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
Assertion
Ref Expression
lcfls1c (𝐺𝐶 ↔ (𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ,𝑓   𝑄,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem lcfls1c
StepHypRef Expression
1 df-3an 1086 . 2 ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
2 lcfls1.c . . 3 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
32lcfls1lem 38823 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
4 lcfls1c.c . . . 4 𝐷 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
54lcfl1lem 38780 . . 3 (𝐺𝐷 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
65anbi1i 626 . 2 ((𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
71, 3, 63bitr4i 306 1 (𝐺𝐶 ↔ (𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  {crab 3113  wss 3884  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336
This theorem is referenced by:  lclkrslem1  38826  lclkrslem2  38827
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