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Theorem lcfls1lem 35841
 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 6103 . . . . . . 7 (𝑓 = 𝐺 → (𝐿𝑓) = (𝐿𝐺))
21fveq2d 6107 . . . . . 6 (𝑓 = 𝐺 → ( ‘(𝐿𝑓)) = ( ‘(𝐿𝐺)))
32fveq2d 6107 . . . . 5 (𝑓 = 𝐺 → ( ‘( ‘(𝐿𝑓))) = ( ‘( ‘(𝐿𝐺))))
43, 1eqeq12d 2625 . . . 4 (𝑓 = 𝐺 → (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
52sseq1d 3595 . . . 4 (𝑓 = 𝐺 → (( ‘(𝐿𝑓)) ⊆ 𝑄 ↔ ( ‘(𝐿𝐺)) ⊆ 𝑄))
64, 5anbi12d 743 . . 3 (𝑓 = 𝐺 → ((( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄) ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
7 lcfls1.c . . 3 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
86, 7elrab2 3333 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
9 3anass 1035 . 2 ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ (𝐺𝐹 ∧ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))
108, 9bitr4i 266 1 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  lcfls1N  35842  lcfls1c  35843  lclkrslem1  35844  lclkrslem2  35845  lclkrs  35846
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