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Theorem lcfls1lem 41517
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 6907 . . . . . . 7 (𝑓 = 𝐺 → (𝐿𝑓) = (𝐿𝐺))
21fveq2d 6911 . . . . . 6 (𝑓 = 𝐺 → ( ‘(𝐿𝑓)) = ( ‘(𝐿𝐺)))
32fveq2d 6911 . . . . 5 (𝑓 = 𝐺 → ( ‘( ‘(𝐿𝑓))) = ( ‘( ‘(𝐿𝐺))))
43, 1eqeq12d 2751 . . . 4 (𝑓 = 𝐺 → (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
52sseq1d 4027 . . . 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 3698 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
9 3anass 1094 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  wss 3963  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:  lcfls1N  41518  lcfls1c  41519  lclkrslem1  41520  lclkrslem2  41521  lclkrs  41522
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