Theorem fcfneii 23188

Description: A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
fcfneii	⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)

Proof of Theorem fcfneii

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

Step	Hyp	Ref	Expression
1		fcfnei 23186	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
2		ineq1 4139	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑠)))
3	2	neeq1d 3003	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅))
4		imaeq2 5965	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝐹 “ 𝑠) = (𝐹 “ 𝑆))
5	4	ineq2d 4146	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑁 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑆)))
6	5	neeq1d 3003	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
7	3, 6	rspc2v 3570	. . . . . 6 ⊢ ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
8	7	ex 413	. . . . 5 ⊢ (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))
9	8	com3r 87	. . . 4 ⊢ (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))
10	9	adantl 482	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))
11	1, 10	syl6bi 252	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))))
12	11	3imp2 1348	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ∩ cin 3886  ∅c0 4256  {csn 4561   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  TopOnctopon 22059  neicnei 22248  Filcfil 22996   fClusf cfcf 23088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-fbas 20594  df-fg 20595  df-top 22043  df-topon 22060  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-fil 22997  df-fm 23089  df-fcls 23092  df-fcf 23093

This theorem is referenced by: (None)