Theorem fcfneii 24066

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 24064	. . 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 4234	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑠)))
3	2	neeq1d 3006	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅))
4		imaeq2 6085	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝐹 “ 𝑠) = (𝐹 “ 𝑆))
5	4	ineq2d 4241	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑁 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑆)))
6	5	neeq1d 3006	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
7	3, 6	rspc2v 3646	. . . . . 6 ⊢ ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
8	7	ex 412	. . . . 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 481	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))
11	1, 10	biimtrdi 253	. 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 1349	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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   ∩ cin 3975  ∅c0 4352  {csn 4648   “ cima 5703  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  TopOnctopon 22937  neicnei 23126  Filcfil 23874   fClusf cfcf 23966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-fil 23875  df-fm 23967  df-fcls 23970  df-fcf 23971

This theorem is referenced by: (None)