Theorem fclsfil 24242

Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
fclsval.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
fclsfil	⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Proof of Theorem fclsfil

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fclsval.x	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	isfcls 24241	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
3	2	simp2bi 1164	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∪ cuni 4870  ‘cfv 6537  (class class class)co 7417  Topctop 23124  clsccl 23249  Filcfil 24077   fClus cfcls 24168

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

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-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-fbas 21588  df-fil 24078  df-fcls 24173

This theorem is used by:  fclstopon  24244  fclsopni  24247  fclselbas  24248  fclsfnflim  24259  cnpfcfi  24272