Theorem cnpfcf 24103

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 23312	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1132	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1183	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		topontop 22975	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5		cnpfcfi 24102	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
6	5	3com23 1140	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
7	6	3expia 1135	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
8	4, 7	sylan 589	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
9	8	ralrimivw 3160	. . . . 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 1201	. . . 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 519	. . 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 416	. 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 786	. . . . . . . . . . . . . 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 23910	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . 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 24020	. . . . . . . . . . . 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 3127	. . . . . . . . . . . . 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 24088	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 fLim 𝑓) ⊆ (𝐽 fClus 𝑓)
22		simpll1 1227	. . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . 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 24034	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
27	23, 24, 25, 26	syl3anc 1392	. . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . . 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 24095	. . . . . . . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 3165	. . . . . . . . . . . . 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 704	. . . . . . . . . . 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 460	. . . . . . . . . 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 3164	. . . . . . . 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 22988	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 24006	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑔 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
57	53, 55, 34, 56	syl3anc 1392	. . . . . . . . . . 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 22976	. . . . . . . . . . . . 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 6873	. . . . . . . . . . 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 2866	. . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
63	62	flimfnfcls 24090	. . . . . . . . . 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 24052	. . . . . . . . . . 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 1392	. . . . . . . . . 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 2850	. . . . . . . . 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 3322	. . . . . . . . 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 313	. . . . . . . 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 261	. . . . . . 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 460	. . . . . 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 3164	. . . 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 579	. . 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 24063	. . 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 261	. 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 214	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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   ⊆ wss 3906  ∪ cuni 4867  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  fBascfbas 21414  Topctop 22955  TopOnctopon 22972   CnP ccnp 23287  Filcfil 23907   FilMap cfm 23995   fLim cflim 23996   fLimf cflf 23997   fClus cfcls 23998   fClusf cfcf 23999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-1o 8439  df-2o 8440  df-map 8812  df-en 8930  df-fin 8933  df-fi 9359  df-fbas 21423  df-fg 21424  df-top 22956  df-topon 22973  df-cld 23081  df-ntr 23082  df-cls 23083  df-nei 23160  df-cnp 23290  df-fil 23908  df-fm 24000  df-flim 24001  df-flf 24002  df-fcls 24003  df-fcf 24004

This theorem is referenced by:  cnfcf  24104