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Theorem lcfl1lem 41474
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 6907 . . . . 5 (𝑓 = 𝐺 → (𝐿𝑓) = (𝐿𝐺))
21fveq2d 6911 . . . 4 (𝑓 = 𝐺 → ( ‘(𝐿𝑓)) = ( ‘(𝐿𝐺)))
32fveq2d 6911 . . 3 (𝑓 = 𝐺 → ( ‘( ‘(𝐿𝑓))) = ( ‘( ‘(𝐿𝐺))))
43, 1eqeq12d 2751 . 2 (𝑓 = 𝐺 → (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
5 lcfl1.c . 2 𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
64, 5elrab2 3698 1 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
Colors of variables: wff setvar class
Syntax hints:  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:  lcfl1  41475  lcfl8b  41487  lclkrlem1  41489  lclkrlem2  41515  lclkr  41516  lcfls1c  41519  lcfrlem9  41533  mapdvalc  41612  mapdval2N  41613  mapdval4N  41615  mapdordlem1a  41617  mapdordlem1bN  41618  mapdrvallem2  41628
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