Theorem fcfneii 24205

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 24203	. . 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 4165	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑠)))
3	2	neeq1d 3016	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅))
4		imaeq2 6057	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝐹 “ 𝑠) = (𝐹 “ 𝑆))
5	4	ineq2d 4172	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑁 ∩ (𝐹 “ 𝑠)) = (𝑁 ∩ (𝐹 “ 𝑆)))
6	5	neeq1d 3016	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑁 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
7	3, 6	rspc2v 3591	. . . . . 6 ⊢ ((𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅))
8	7	ex 417	. . . . 5 ⊢ (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))
9	8	com3r 88	. . . 4 ⊢ (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑁 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑆 ∈ 𝐿 → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)))
10	9	adantl 486	. . 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 256	. 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 1367	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078   ∩ cin 3903  ∅c0 4285  {csn 4588   “ cima 5663  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412  TopOnctopon 23078  neicnei 23265  Filcfil 24013   fClusf cfcf 24105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-fbas 21530  df-fg 21531  df-top 23062  df-topon 23079  df-cld 23187  df-ntr 23188  df-cls 23189  df-nei 23266  df-fil 24014  df-fm 24106  df-fcls 24109  df-fcf 24110

This theorem is used by: (None)