Theorem cnpfcf 23255

Description: A function 𝐹 is continuous at point 𝐴 iff 𝐹 respects cluster points there. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
cnpfcf	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐹   𝑓,𝐽   𝑓,𝐾   𝑓,𝑋   𝑓,𝑌

Proof of Theorem cnpfcf

Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnpf2 22464	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1117	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1167	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		topontop 22125	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5		cnpfcfi 23254	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
6	5	3com23 1125	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
7	6	3expia 1120	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
8	4, 7	sylan 580	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
9	8	ralrimivw 3142	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
10	9	3ad2antl2 1185	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
11	3, 10	jca 512	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
12	11	ex 413	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
13		simplrl 774	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → 𝑔 ∈ (Fil‘𝑋))
14		filfbas 23062	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ (fBas‘𝑋))
15	13, 14	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → 𝑔 ∈ (fBas‘𝑋))
16		simprl 768	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → ℎ ∈ (Fil‘𝑌))
17		simpllr 773	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → 𝐹:𝑋⟶𝑌)
18		simprr 770	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)
19	15, 16, 17, 18	fmfnfm 23172	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → ∃𝑓 ∈ (Fil‘𝑋)(𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))
20		r19.29 3112	. . . . . . . . . . . . 13 ⊢ ((∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ ∃𝑓 ∈ (Fil‘𝑋)(𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓))) → ∃𝑓 ∈ (Fil‘𝑋)((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓))))
21		flimfcls 23240	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 fLim 𝑓) ⊆ (𝐽 fClus 𝑓)
22		simpll1 1211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐽 ∈ (TopOn‘𝑋))
23	22	ad2antrr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐽 ∈ (TopOn‘𝑋))
24		simprl 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
25		simprrl 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝑔 ⊆ 𝑓)
26		flimss2 23186	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
27	23, 24, 25, 26	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
28		simprr 770	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐴 ∈ (𝐽 fLim 𝑔))
29	28	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐴 ∈ (𝐽 fLim 𝑔))
30	27, 29	sseldd 3926	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐴 ∈ (𝐽 fLim 𝑓))
31	21, 30	sselid 3923	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐴 ∈ (𝐽 fClus 𝑓))
32		simpll2 1212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐾 ∈ (TopOn‘𝑌))
33	32	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐾 ∈ (TopOn‘𝑌))
34		simplr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐹:𝑋⟶𝑌)
35	34	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐹:𝑋⟶𝑌)
36		fcfval 23247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
37	33, 24, 35, 36	syl3anc 1370	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
38		simprrr 779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → ℎ = ((𝑌 FilMap 𝐹)‘𝑓))
39	38	oveq2d 7324	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → (𝐾 fClus ℎ) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
40	37, 39	eqtr4d 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ℎ))
41	40	eleq2d 2821	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹) ↔ (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
42	41	biimpd 228	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
43	31, 42	embantd 59	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
44	43	expr 457	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ 𝑓 ∈ (Fil‘𝑋)) → ((𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓)) → ((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
45	44	impcomd 412	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) ∧ 𝑓 ∈ (Fil‘𝑋)) → (((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓))) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
46	45	rexlimdva 3147	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → (∃𝑓 ∈ (Fil‘𝑋)((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ (𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓))) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
47	20, 46	syl5 34	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → ((∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ ∃𝑓 ∈ (Fil‘𝑋)(𝑔 ⊆ 𝑓 ∧ ℎ = ((𝑌 FilMap 𝐹)‘𝑓))) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
48	19, 47	mpan2d 691	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (ℎ ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)))
49	48	expr 457	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ℎ ∈ (Fil‘𝑌)) → (((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
50	49	com23 86	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ℎ ∈ (Fil‘𝑌)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
51	50	ralrimdva 3146	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → ∀ℎ ∈ (Fil‘𝑌)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
52		toponmax 22138	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
53	32, 52	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑌 ∈ 𝐾)
54		simprl 768	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑔 ∈ (Fil‘𝑋))
55	54, 14	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑔 ∈ (fBas‘𝑋))
56		fmfil 23158	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑔 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
57	53, 55, 34, 56	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
58		toponuni 22126	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
59	32, 58	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑌 = ∪ 𝐾)
60	59	fveq2d 6808	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (Fil‘𝑌) = (Fil‘∪ 𝐾))
61	57, 60	eleqtrd 2838	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘∪ 𝐾))
62		eqid 2735	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
63	62	flimfnfcls 23242	. . . . . . . . . 10 ⊢ (((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘∪ 𝐾) → ((𝐹‘𝐴) ∈ (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)) ↔ ∀ℎ ∈ (Fil‘∪ 𝐾)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
64	61, 63	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐹‘𝐴) ∈ (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)) ↔ ∀ℎ ∈ (Fil‘∪ 𝐾)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
65		flfval 23204	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝐾 fLimf 𝑔)‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)))
66	32, 54, 34, 65	syl3anc 1370	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐾 fLimf 𝑔)‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)))
67	66	eleq2d 2821	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹) ↔ (𝐹‘𝐴) ∈ (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔))))
68	60	raleqdv 3308	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (∀ℎ ∈ (Fil‘𝑌)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ)) ↔ ∀ℎ ∈ (Fil‘∪ 𝐾)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
69	64, 67, 68	3bitr4d 310	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹) ↔ ∀ℎ ∈ (Fil‘𝑌)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ ℎ → (𝐹‘𝐴) ∈ (𝐾 fClus ℎ))))
70	51, 69	sylibrd 258	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹)))
71	70	expr 457	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝑔) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹))))
72	71	com23 86	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹))))
73	72	ralrimdva 3146	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → ∀𝑔 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹))))
74	73	imdistanda 572	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑔 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹)))))
75		cnpflf 23215	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑔 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹)))))
76	74, 75	sylibrd 258	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
77	12, 76	impbid 211	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1538   ∈ wcel 2103  ∀wral 3060  ∃wrex 3069   ⊆ wss 3891  ∪ cuni 4843  ⟶wf 6454  ‘cfv 6458  (class class class)co 7308  fBascfbas 20648  Topctop 22105  TopOnctopon 22122   CnP ccnp 22439  Filcfil 23059   FilMap cfm 23147   fLim cflim 23148   fLimf cflf 23149   fClus cfcls 23150   fClusf cfcf 23151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1968  ax-7 2008  ax-8 2105  ax-9 2113  ax-10 2134  ax-11 2151  ax-12 2168  ax-ext 2706  ax-rep 5217  ax-sep 5231  ax-nul 5238  ax-pow 5296  ax-pr 5360  ax-un 7621

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2727  df-clel 2813  df-nfc 2885  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3340  df-rab 3357  df-v 3438  df-sbc 3721  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4844  df-int 4886  df-iun 4932  df-iin 4933  df-br 5081  df-opab 5143  df-mpt 5164  df-tr 5198  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7311  df-oprab 7312  df-mpo 7313  df-om 7749  df-1st 7867  df-2nd 7868  df-1o 8332  df-er 8534  df-map 8653  df-en 8770  df-fin 8773  df-fi 9228  df-fbas 20657  df-fg 20658  df-top 22106  df-topon 22123  df-cld 22233  df-ntr 22234  df-cls 22235  df-nei 22312  df-cnp 22442  df-fil 23060  df-fm 23152  df-flim 23153  df-flf 23154  df-fcls 23155  df-fcf 23156

This theorem is referenced by:  cnfcf  23256