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Metamath Proof Explorer

Theorem cnfcf

Description: Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009) (Revised by Stefan O'Rear, 9-Aug-2015)

Ref Expression
Assertion cnfcf ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )

Proof

Step Hyp Ref Expression
1 cncnp ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑋 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ) ) )
2 simplr ⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) ∧ 𝑥 ∈ 𝑋 ) → 𝐹 : 𝑋 ⟶ 𝑌 )
3 cnpfcf ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) ) )
4 3 ad4ant124 ⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) ) )
5 2 4 mpbirand ⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ↔ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
6 5 ralbidva ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑥 ∈ 𝑋 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
7 ralcom ⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ↔ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ 𝑋 ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) )
8 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
9 8 fclselbas ⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → 𝑥 ∈ ∪ 𝐽 )
10 toponuni ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
11 10 ad2antrr ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → 𝑋 = ∪ 𝐽 )
12 11 eleq2d ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( 𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽 ) )
13 9 12 imbitrrid ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → 𝑥 ∈ 𝑋 ) )
14 13 pm4.71rd ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ↔ ( 𝑥 ∈ 𝑋 ∧ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ) ) )
15 14 imbi1d ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ↔ ( ( 𝑥 ∈ 𝑋 ∧ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
16 impexp ⊢ ( ( ( 𝑥 ∈ 𝑋 ∧ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ↔ ( 𝑥 ∈ 𝑋 → ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
17 15 16 bitrdi ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ↔ ( 𝑥 ∈ 𝑋 → ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) ) )
18 17 ralbidv2 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ↔ ∀ 𝑥 ∈ 𝑋 ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
19 18 ralbidv ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ↔ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ 𝑋 ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
20 7 19 bitr4id ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝑥 ∈ ( 𝐽 fClus 𝑓 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ↔ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) )
21 6 20 bitrd ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑥 ∈ 𝑋 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ↔ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) )
22 21 pm5.32da ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑋 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑥 ) ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )
23 1 22 bitrd ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ∀ 𝑥 ∈ ( 𝐽 fClus 𝑓 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fClusf 𝑓 ) ‘ 𝐹 ) ) ) )