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Theorem fcfnei 21762
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 21761 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))
2 simpll1 1098 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘𝑋))
3 topontop 20650 . . . . . . . 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 477 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ∈ ((nei‘𝐽)‘{𝐴}))
6 eqid 2621 . . . . . . . . 9 𝐽 = 𝐽
76neii1 20833 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
84, 5, 7syl2anc 692 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
96ntrss2 20784 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑛 𝐽) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
104, 8, 9syl2anc 692 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
11 simplr 791 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
12 toponuni 20651 . . . . . . . . . . . . 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 2700 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
1514snssd 4314 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
166neiint 20831 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑛 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
174, 15, 8, 16syl3anc 1323 . . . . . . . . 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 4301 . . . . . . . . 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 20776 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑛 𝐽) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
234, 8, 22syl2anc 692 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
24 eleq2 2687 . . . . . . . . . 10 (𝑜 = ((int‘𝐽)‘𝑛) → (𝐴𝑜𝐴 ∈ ((int‘𝐽)‘𝑛)))
25 ineq1 3790 . . . . . . . . . . . 12 (𝑜 = ((int‘𝐽)‘𝑛) → (𝑜 ∩ (𝐹𝑠)) = (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)))
2625neeq1d 2849 . . . . . . . . . . 11 (𝑜 = ((int‘𝐽)‘𝑛) → ((𝑜 ∩ (𝐹𝑠)) ≠ ∅ ↔ (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
2726ralbidv 2981 . . . . . . . . . 10 (𝑜 = ((int‘𝐽)‘𝑛) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ ↔ ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅))
2824, 27imbi12d 334 . . . . . . . . 9 (𝑜 = ((int‘𝐽)‘𝑛) → ((𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅)))
2928rspcv 3294 . . . . . . . 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 3821 . . . . . . . 8 (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)))
33 ssn0 3953 . . . . . . . . 9 (((((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ⊆ (𝑛 ∩ (𝐹𝑠)) ∧ (((int‘𝐽)‘𝑛) ∩ (𝐹𝑠)) ≠ ∅) → (𝑛 ∩ (𝐹𝑠)) ≠ ∅)
3433ex 450 . . . . . . . 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 2958 . . . . . 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 2964 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅))
39 simpl1 1062 . . . . . . . . . 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 20846 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽𝐴𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
42413expb 1263 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
4340, 42sylan 488 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑜𝐽𝐴𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
44 ineq1 3790 . . . . . . . . . . 11 (𝑛 = 𝑜 → (𝑛 ∩ (𝐹𝑠)) = (𝑜 ∩ (𝐹𝑠)))
4544neeq1d 2849 . . . . . . . . . 10 (𝑛 = 𝑜 → ((𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4645ralbidv 2981 . . . . . . . . 9 (𝑛 = 𝑜 → (∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4746rspcv 3294 . . . . . . . 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 642 . . . . . 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 2964 . . . 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 672 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cin 3558  wss 3559  c0 3896  {csn 4153   cuni 4407  cima 5082  wf 5848  cfv 5852  (class class class)co 6610  Topctop 20630  TopOnctopon 20647  intcnt 20744  neicnei 20824  Filcfil 21572   fClusf cfcf 21664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-fbas 19675  df-fg 19676  df-top 20631  df-topon 20648  df-cld 20746  df-ntr 20747  df-cls 20748  df-nei 20825  df-fil 21573  df-fm 21665  df-fcls 21668  df-fcf 21669
This theorem is referenced by:  fcfneii  21764
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