Theorem lcfls1lem 39285

Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)

Hypothesis

Ref	Expression
lcfls1.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}

Assertion

Ref	Expression
lcfls1lem	⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))

Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   ⊥ ,𝑓   𝑄,𝑓

Allowed substitution hint:   𝐶(𝑓)

Proof of Theorem lcfls1lem

Step	Hyp	Ref	Expression
1		fveq2 6717	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝐿‘𝑓) = (𝐿‘𝐺))
2	1	fveq2d 6721	. . . . . 6 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘(𝐿‘𝑓)) = ( ⊥ ‘(𝐿‘𝐺)))
3	2	fveq2d 6721	. . . . 5 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
4	3, 1	eqeq12d 2753	. . . 4 ⊢ (𝑓 = 𝐺 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
5	2	sseq1d 3932	. . . 4 ⊢ (𝑓 = 𝐺 → (( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄 ↔ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
6	4, 5	anbi12d 634	. . 3 ⊢ (𝑓 = 𝐺 → ((( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄) ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)))
7		lcfls1.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}
8	6, 7	elrab2 3605	. 2 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)))
9		3anass 1097	. 2 ⊢ ((𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ (𝐺 ∈ 𝐹 ∧ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)))
10	8, 9	bitr4i 281	1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {crab 3065   ⊆ wss 3866  ‘cfv 6380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388

This theorem is referenced by:  lcfls1N  39286  lcfls1c  39287  lclkrslem1  39288  lclkrslem2  39289  lclkrs  39290