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Theorem fcfnei 22164
Description: The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
fcfnei ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
Distinct variable groups:   𝐴,𝑛   𝑛,𝑠,𝐽   𝑛,𝐿,𝑠   𝑛,𝐹,𝑠   𝑛,𝑋,𝑠   𝑛,𝑌,𝑠
Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem fcfnei
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 isfcf 22163 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))
2 simpll1 1270 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘𝑋))
3 topontop 21043 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
42, 3syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
5 simpr 478 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ∈ ((nei‘𝐽)‘{𝐴}))
6 eqid 2797 . . . . . . . . 9 𝐽 = 𝐽
76neii1 21236 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
84, 5, 7syl2anc 580 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
96ntrss2 21187 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛 𝐽) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
104, 8, 9syl2anc 580 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
11 simplr 786 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
12 toponuni 21044 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
132, 12syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = 𝐽)
1411, 13eleqtrd 2878 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
1514snssd 4526 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
166neiint 21234 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑛 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
174, 15, 8, 16syl3anc 1491 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
185, 17mpbid 224 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑛))
19 snssg 4501 . . . . . . . . 9 (𝐴𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑛) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
2011, 19syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝐴 ∈ ((int‘𝐽)‘𝑛) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
2118, 20mpbird 249 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑛))
226ntropn 21179 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑛 𝐽) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
234, 8, 22syl2anc 580 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
24 eleq2 2865 . . . . . . . . . 10 (𝑜 = ((int‘𝐽)‘𝑛) → (𝐴𝑜𝐴 ∈ ((int‘𝐽)‘𝑛)))
25 ineq1 4003 . . . . . . . . . . . 12 (𝑜 = ((int‘𝐽)‘𝑛) → (𝑜 ∩ (𝐹𝑠)) = (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)))
2625neeq1d 3028 . . . . . . . . . . 11 (𝑜 = ((int‘𝐽)‘𝑛) → ((𝑜 ∩ (𝐹𝑠)) ≠ ∅ ↔ (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
2726ralbidv 3165 . . . . . . . . . 10 (𝑜 = ((int‘𝐽)‘𝑛) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ ↔ ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
2824, 27imbi12d 336 . . . . . . . . 9 (𝑜 = ((int‘𝐽)‘𝑛) → ((𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅)))
2928rspcv 3491 . . . . . . . 8 (((int‘𝐽)‘𝑛) ∈ 𝐽 → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅)))
3023, 29syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅)))
3121, 30mpid 44 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
32 ssrin 4031 . . . . . . . 8 (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)))
33 ssn0 4170 . . . . . . . . 9 (((((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)) ∧ (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅) → (𝑛 ∩ (𝐹𝑠)) ≠ ∅)
3433ex 402 . . . . . . . 8 ((((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)) → ((((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅ → (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
3532, 34syl 17 . . . . . . 7 (((int‘𝐽)‘𝑛) ⊆ 𝑛 → ((((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅ → (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
3635ralimdv 3142 . . . . . 6 (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
3710, 31, 36sylsyld 61 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
3837ralrimdva 3148 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
39 simpl1 1243 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4039, 3syl 17 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
41 opnneip 21249 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽𝐴𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
42413expb 1150 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
4340, 42sylan 576 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
44 ineq1 4003 . . . . . . . . . . 11 (𝑛 = 𝑜 → (𝑛 ∩ (𝐹𝑠)) = (𝑜 ∩ (𝐹𝑠)))
4544neeq1d 3028 . . . . . . . . . 10 (𝑛 = 𝑜 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4645ralbidv 3165 . . . . . . . . 9 (𝑛 = 𝑜 → (∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4746rspcv 3491 . . . . . . . 8 (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4843, 47syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4948expr 449 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴𝑜 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
5049com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
5150ralrimdva 3148 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
5238, 51impbid 204 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
5352pm5.32da 575 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
541, 53bitrd 271 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2969  wral 3087  cin 3766  wss 3767  c0 4113  {csn 4366   cuni 4626  cima 5313  wf 6095  cfv 6099  (class class class)co 6876  Topctop 21023  TopOnctopon 21040  intcnt 21147  neicnei 21227  Filcfil 21974   fClusf cfcf 22066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-map 8095  df-fbas 20062  df-fg 20063  df-top 21024  df-topon 21041  df-cld 21149  df-ntr 21150  df-cls 21151  df-nei 21228  df-fil 21975  df-fm 22067  df-fcls 22070  df-fcf 22071
This theorem is referenced by:  fcfneii  22166
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