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Theorem lcfls1lem 40405
Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
Hypothesis
Ref Expression
lcfls1.c 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
Assertion
Ref Expression
lcfls1lem (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ,𝑓   𝑄,𝑓
Allowed substitution hint:   𝐶(𝑓)

Proof of Theorem lcfls1lem
StepHypRef Expression
1 fveq2 6892 . . . . . . 7 (𝑓 = 𝐺 → (𝐿𝑓) = (𝐿𝐺))
21fveq2d 6896 . . . . . 6 (𝑓 = 𝐺 → ( ‘(𝐿𝑓)) = ( ‘(𝐿𝐺)))
32fveq2d 6896 . . . . 5 (𝑓 = 𝐺 → ( ‘( ‘(𝐿𝑓))) = ( ‘( ‘(𝐿𝐺))))
43, 1eqeq12d 2749 . . . 4 (𝑓 = 𝐺 → (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
52sseq1d 4014 . . . 4 (𝑓 = 𝐺 → (( ‘(𝐿𝑓)) ⊆ 𝑄 ↔ ( ‘(𝐿𝐺)) ⊆ 𝑄))
64, 5anbi12d 632 . . 3 (𝑓 = 𝐺 → ((( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄) ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
7 lcfls1.c . . 3 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
86, 7elrab2 3687 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
9 3anass 1096 . 2 ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ (𝐺𝐹 ∧ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
108, 9bitr4i 278 1 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3433  wss 3949  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552
This theorem is referenced by:  lcfls1N  40406  lcfls1c  40407  lclkrslem1  40408  lclkrslem2  40409  lclkrs  40410
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