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Theorem lcfls1c 38552
Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
Hypotheses
Ref Expression
lcfls1.c 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
lcfls1c.c 𝐷 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
Assertion
Ref Expression
lcfls1c (𝐺𝐶 ↔ (𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ,𝑓   𝑄,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem lcfls1c
StepHypRef Expression
1 df-3an 1081 . 2 ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
2 lcfls1.c . . 3 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
32lcfls1lem 38550 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
4 lcfls1c.c . . . 4 𝐷 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
54lcfl1lem 38507 . . 3 (𝐺𝐷 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
65anbi1i 623 . 2 ((𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
71, 3, 63bitr4i 304 1 (𝐺𝐶 ↔ (𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  wss 3933  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  lclkrslem1  38553  lclkrslem2  38554
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