Theorem lcfl1lem 35798

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 6103	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝐿‘𝑓) = (𝐿‘𝐺))
2	1	fveq2d 6107	. . . 4 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘(𝐿‘𝑓)) = ( ⊥ ‘(𝐿‘𝐺)))
3	2	fveq2d 6107	. . 3 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
4	3, 1	eqeq12d 2625	. 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 3333	1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘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:  lcfl1  35799  lcfl8b  35811  lclkrlem1  35813  lclkrlem2  35839  lclkr  35840  lcfls1c  35843  lcfrlem9  35857  mapdvalc  35936  mapdval2N  35937  mapdval4N  35939  mapdordlem1a  35941  mapdordlem1bN  35942  mapdrvallem2  35952