Theorem cnpfcf 22651

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 21860	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1114	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1164	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		topontop 21523	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5		cnpfcfi 22650	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
6	5	3com23 1122	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
7	6	3expia 1117	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
8	4, 7	sylan 582	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
9	8	ralrimivw 3185	. . . . 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 1182	. . . 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 514	. . 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 415	. 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 775	. . . . . . . . . . . . . 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 22458	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 22568	. . . . . . . . . . . 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 3256	. . . . . . . . . . . . 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 22636	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 fLim 𝑓) ⊆ (𝐽 fClus 𝑓)
22		simpll1 1208	. . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . 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 22582	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
27	23, 24, 25, 26	syl3anc 1367	. . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . . . . 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	sseldi 3967	. . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . 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 22643	. . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . 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 7174	. . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . 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 2900	. . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . 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 3286	. . . . . . . . . . . . 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 692	. . . . . . . . . . 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 459	. . . . . . . . . 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 3191	. . . . . . . 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 21536	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 22554	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑔 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
57	53, 55, 34, 56	syl3anc 1367	. . . . . . . . . . 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 21524	. . . . . . . . . . . . 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 6676	. . . . . . . . . . 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 2917	. . . . . . . . . 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 2823	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
63	62	flimfnfcls 22638	. . . . . . . . . 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 22600	. . . . . . . . . . 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 1367	. . . . . . . . . 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 2900	. . . . . . . . 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 3417	. . . . . . . . 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 459	. . . . . 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 3191	. . . 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 574	. . 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 22611	. . 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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938  ∪ cuni 4840  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  fBascfbas 20535  Topctop 21503  TopOnctopon 21520   CnP ccnp 21835  Filcfil 22455   FilMap cfm 22543   fLim cflim 22544   fLimf cflf 22545   fClus cfcls 22546   fClusf cfcf 22547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-fin 8515  df-fi 8877  df-fbas 20544  df-fg 20545  df-top 21504  df-topon 21521  df-cld 21629  df-ntr 21630  df-cls 21631  df-nei 21708  df-cnp 21838  df-fil 22456  df-fm 22548  df-flim 22549  df-flf 22550  df-fcls 22551  df-fcf 22552

This theorem is referenced by:  cnfcf  22652