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Theorem lcfl1 41475
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.c . . 3 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
21lcfl1lem 41474 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
3 lcfl1.g . . 3 (𝜑𝐺𝐹)
43biantrurd 532 . 2 (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺))))
52, 4bitr4id 290 1 (𝜑 → (𝐺𝐶 ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  lcfl2  41476  lcfl5  41479  lcfl5a  41480  lcfl6  41483  lcfl8  41485  lcfl8a  41486  lclkrlem2  41515
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