Theorem fclssscls 23997

Description: The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclssscls	⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem fclssscls

Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isfcls 23988	. . . 4 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)))
3	2	simp3bi 1148	. . 3 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) → ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠))
4		fveq2 6836	. . . . 5 ⊢ (𝑠 = 𝑆 → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝑆))
5	4	eleq2d 2823	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘𝑠) ↔ 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
6	5	rspcv 3561	. . 3 ⊢ (𝑆 ∈ 𝐹 → (∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
7	3, 6	syl5 34	. 2 ⊢ (𝑆 ∈ 𝐹 → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
8	7	ssrdv 3928	1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ∪ cuni 4851  ‘cfv 6494  (class class class)co 7362  Topctop 22872  clsccl 22997  Filcfil 23824   fClus cfcls 23915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-fbas 21345  df-fil 23825  df-fcls 23920

This theorem is referenced by:  fclscmp  24009