Theorem fcfneii 21841

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 21839	. . 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 3807	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑠)))
3	2	neeq1d 2853	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅))
4		imaeq2 5462	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝐹 “ 𝑠) = (𝐹 “ 𝑆))
5	4	ineq2d 3814	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑁 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑆)))
6	5	neeq1d 2853	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
7	3, 6	rspc2v 3322	. . . . . 6 ⊢ ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
8	7	ex 450	. . . . 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 243	. 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 1282	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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912   ∩ cin 3573  ∅c0 3915  {csn 4177   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  TopOnctopon 20715  neicnei 20901  Filcfil 21649   fClusf cfcf 21741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650  df-fm 21742  df-fcls 21745  df-fcf 21746

This theorem is referenced by: (None)