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Theorem lcfls1c 37327
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 1074 . 2 ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
2 lcfls1.c . . 3 𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}
32lcfls1lem 37325 . 2 (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
4 lcfls1c.c . . . 4 𝐷 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
54lcfl1lem 37282 . . 3 (𝐺𝐷 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))
65anbi1i 733 . 2 ((𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄) ↔ ((𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
71, 3, 63bitr4i 292 1 (𝐺𝐶 ↔ (𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  wss 3715  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  lclkrslem1  37328  lclkrslem2  37329
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