Theorem lcfl1lem 42751

Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)

Hypothesis

Ref	Expression
lcfl1.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}

Assertion

Ref	Expression
lcfl1lem	⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))

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

Allowed substitution hint:   𝐶(𝑓)

Proof of Theorem lcfl1lem

Step	Hyp	Ref	Expression
1		fveq2 6911	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝐿‘𝑓) = (𝐿‘𝐺))
2	1	fveq2d 6915	. . . 4 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘(𝐿‘𝑓)) = ( ⊥ ‘(𝐿‘𝐺)))
3	2	fveq2d 6915	. . 3 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
4	3, 1	eqeq12d 2779	. 2 ⊢ (𝑓 = 𝐺 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
5		lcfl1.c	. 2 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
6	4, 5	elrab2 3652	1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569

This theorem is used by:  lcfl1  42752  lcfl8b  42764  lclkrlem1  42766  lclkrlem2  42792  lclkr  42793  lcfls1c  42796  lcfrlem9  42810  mapdvalc  42889  mapdval2N  42890  mapdval4N  42892  mapdordlem1a  42894  mapdordlem1bN  42895  mapdrvallem2  42905