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Theorem lcfl1lem 38732
Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
Hypothesis
Ref Expression
lcfl1.c 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
Assertion
Ref Expression
lcfl1lem (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ,𝑓
Allowed substitution hint:   𝐶(𝑓)

Proof of Theorem lcfl1lem
StepHypRef Expression
1 fveq2 6661 . . . . 5 (𝑓 = 𝐺 → (𝐿𝑓) = (𝐿𝐺))
21fveq2d 6665 . . . 4 (𝑓 = 𝐺 → ( ‘(𝐿𝑓)) = ( ‘(𝐿𝐺)))
32fveq2d 6665 . . 3 (𝑓 = 𝐺 → ( ‘( ‘(𝐿𝑓))) = ( ‘( ‘(𝐿𝐺))))
43, 1eqeq12d 2840 . 2 (𝑓 = 𝐺 → (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
5 lcfl1.c . 2 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
64, 5elrab2 3669 1 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  {crab 3137  cfv 6343
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351
This theorem is referenced by:  lcfl1  38733  lcfl8b  38745  lclkrlem1  38747  lclkrlem2  38773  lclkr  38774  lcfls1c  38777  lcfrlem9  38791  mapdvalc  38870  mapdval2N  38871  mapdval4N  38873  mapdordlem1a  38875  mapdordlem1bN  38876  mapdrvallem2  38886
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