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Theorem lcfl1 40666
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 40665 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
3 lcfl1.g . . 3 (𝜑𝐺𝐹)
43biantrurd 531 . 2 (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺))))
52, 4bitr4id 289 1 (𝜑 → (𝐺𝐶 ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  {crab 3430  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550
This theorem is referenced by:  lcfl2  40667  lcfl5  40670  lcfl5a  40671  lcfl6  40674  lcfl8  40676  lcfl8a  40677  lclkrlem2  40706
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