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Theorem lcfl1 35799
 Description: Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
Hypotheses
Ref Expression
lcfl1.c 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
lcfl1.g (𝜑𝐺𝐹)
Assertion
Ref Expression
lcfl1 (𝜑 → (𝐺𝐶 ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ,𝑓
Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)

Proof of Theorem lcfl1
StepHypRef Expression
1 lcfl1.g . . 3 (𝜑𝐺𝐹)
21biantrurd 528 . 2 (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺))))
3 lcfl1.c . . 3 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
43lcfl1lem 35798 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
52, 4syl6rbbr 278 1 (𝜑 → (𝐺𝐶 ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  lcfl2  35800  lcfl5  35803  lcfl5a  35804  lcfl6  35807  lcfl8  35809  lcfl8a  35810  lclkrlem2  35839
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