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Theorem fcfnei 22059
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 22058 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))
2 simpll1 1254 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘𝑋))
3 topontop 20938 . . . . . . . 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 471 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ∈ ((nei‘𝐽)‘{𝐴}))
6 eqid 2771 . . . . . . . . 9 𝐽 = 𝐽
76neii1 21131 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
84, 5, 7syl2anc 573 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
96ntrss2 21082 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛 𝐽) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
104, 8, 9syl2anc 573 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
11 simplr 752 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
12 toponuni 20939 . . . . . . . . . . . . 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 2852 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
1514snssd 4475 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
166neiint 21129 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑛 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
174, 15, 8, 16syl3anc 1476 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
185, 17mpbid 222 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑛))
19 snssg 4450 . . . . . . . . 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 247 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑛))
226ntropn 21074 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑛 𝐽) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
234, 8, 22syl2anc 573 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
24 eleq2 2839 . . . . . . . . . 10 (𝑜 = ((int‘𝐽)‘𝑛) → (𝐴𝑜𝐴 ∈ ((int‘𝐽)‘𝑛)))
25 ineq1 3958 . . . . . . . . . . . 12 (𝑜 = ((int‘𝐽)‘𝑛) → (𝑜 ∩ (𝐹𝑠)) = (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)))
2625neeq1d 3002 . . . . . . . . . . 11 (𝑜 = ((int‘𝐽)‘𝑛) → ((𝑜 ∩ (𝐹𝑠)) ≠ ∅ ↔ (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
2726ralbidv 3135 . . . . . . . . . 10 (𝑜 = ((int‘𝐽)‘𝑛) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ ↔ ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
2824, 27imbi12d 333 . . . . . . . . 9 (𝑜 = ((int‘𝐽)‘𝑛) → ((𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅)))
2928rspcv 3456 . . . . . . . 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 3986 . . . . . . . 8 (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)))
33 ssn0 4120 . . . . . . . . 9 (((((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)) ∧ (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅) → (𝑛 ∩ (𝐹𝑠)) ≠ ∅)
3433ex 397 . . . . . . . 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 3112 . . . . . 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 3118 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
39 simpl1 1227 . . . . . . . . . 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 21144 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽𝐴𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
42413expb 1113 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
4340, 42sylan 569 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
44 ineq1 3958 . . . . . . . . . . 11 (𝑛 = 𝑜 → (𝑛 ∩ (𝐹𝑠)) = (𝑜 ∩ (𝐹𝑠)))
4544neeq1d 3002 . . . . . . . . . 10 (𝑛 = 𝑜 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4645ralbidv 3135 . . . . . . . . 9 (𝑛 = 𝑜 → (∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4746rspcv 3456 . . . . . . . 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 444 . . . . . 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 3118 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
5238, 51impbid 202 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
5352pm5.32da 568 . 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 268 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  cin 3722  wss 3723  c0 4063  {csn 4316   cuni 4574  cima 5252  wf 6027  cfv 6031  (class class class)co 6793  Topctop 20918  TopOnctopon 20935  intcnt 21042  neicnei 21122  Filcfil 21869   fClusf cfcf 21961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-fbas 19958  df-fg 19959  df-top 20919  df-topon 20936  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-fil 21870  df-fm 21962  df-fcls 21965  df-fcf 21966
This theorem is referenced by:  fcfneii  22061
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