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Theorem cnpfcf 23939
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) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnpf2 23148 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
213expa 1116 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
323adantl3 1166 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
4 topontop 22809 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5 cnpfcfi 23938 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
653com23 1124 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
763expia 1119 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
84, 7sylan 579 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
98ralrimivw 3146 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
1093ad2antl2 1184 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
113, 10jca 511 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
1211ex 412 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
13 simplrl 776 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → 𝑔 ∈ (Fil‘𝑋))
14 filfbas 23746 . . . . . . . . . . . . . 14 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ (fBas‘𝑋))
1513, 14syl 17 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → 𝑔 ∈ (fBas‘𝑋))
16 simprl 770 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → ∈ (Fil‘𝑌))
17 simpllr 775 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → 𝐹:𝑋𝑌)
18 simprr 772 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )
1915, 16, 17, 18fmfnfm 23856 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → ∃𝑓 ∈ (Fil‘𝑋)(𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))
20 r19.29 3110 . . . . . . . . . . . . 13 ((∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ ∃𝑓 ∈ (Fil‘𝑋)(𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓))) → ∃𝑓 ∈ (Fil‘𝑋)((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓))))
21 flimfcls 23924 . . . . . . . . . . . . . . . . . 18 (𝐽 fLim 𝑓) ⊆ (𝐽 fClus 𝑓)
22 simpll1 1210 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐽 ∈ (TopOn‘𝑋))
2322ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐽 ∈ (TopOn‘𝑋))
24 simprl 770 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
25 simprrl 780 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝑔𝑓)
26 flimss2 23870 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑔𝑓) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
2723, 24, 25, 26syl3anc 1369 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
28 simprr 772 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐴 ∈ (𝐽 fLim 𝑔))
2928ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐴 ∈ (𝐽 fLim 𝑔))
3027, 29sseldd 3980 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐴 ∈ (𝐽 fLim 𝑓))
3121, 30sselid 3977 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐴 ∈ (𝐽 fClus 𝑓))
32 simpll2 1211 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐾 ∈ (TopOn‘𝑌))
3332ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐾 ∈ (TopOn‘𝑌))
34 simplr 768 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐹:𝑋𝑌)
3534ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → 𝐹:𝑋𝑌)
36 fcfval 23931 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
3733, 24, 35, 36syl3anc 1369 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
38 simprrr 781 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → = ((𝑌 FilMap 𝐹)‘𝑓))
3938oveq2d 7431 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → (𝐾 fClus ) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
4037, 39eqtr4d 2771 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ))
4140eleq2d 2815 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹) ↔ (𝐹𝐴) ∈ (𝐾 fClus )))
4241biimpd 228 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹) → (𝐹𝐴) ∈ (𝐾 fClus )))
4331, 42embantd 59 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)))) → ((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹𝐴) ∈ (𝐾 fClus )))
4443expr 456 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ 𝑓 ∈ (Fil‘𝑋)) → ((𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓)) → ((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹𝐴) ∈ (𝐾 fClus ))))
4544impcomd 411 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) ∧ 𝑓 ∈ (Fil‘𝑋)) → (((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓))) → (𝐹𝐴) ∈ (𝐾 fClus )))
4645rexlimdva 3151 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → (∃𝑓 ∈ (Fil‘𝑋)((𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ (𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓))) → (𝐹𝐴) ∈ (𝐾 fClus )))
4720, 46syl5 34 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → ((∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ∧ ∃𝑓 ∈ (Fil‘𝑋)(𝑔𝑓 = ((𝑌 FilMap 𝐹)‘𝑓))) → (𝐹𝐴) ∈ (𝐾 fClus )))
4819, 47mpan2d 693 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ( ∈ (Fil‘𝑌) ∧ ((𝑌 FilMap 𝐹)‘𝑔) ⊆ )) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹𝐴) ∈ (𝐾 fClus )))
4948expr 456 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ∈ (Fil‘𝑌)) → (((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹𝐴) ∈ (𝐾 fClus ))))
5049com23 86 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ ∈ (Fil‘𝑌)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus ))))
5150ralrimdva 3150 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → ∀ ∈ (Fil‘𝑌)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus ))))
52 toponmax 22822 . . . . . . . . . . . . 13 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
5332, 52syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑌𝐾)
54 simprl 770 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑔 ∈ (Fil‘𝑋))
5554, 14syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑔 ∈ (fBas‘𝑋))
56 fmfil 23842 . . . . . . . . . . . 12 ((𝑌𝐾𝑔 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
5753, 55, 34, 56syl3anc 1369 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
58 toponuni 22810 . . . . . . . . . . . . 13 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
5932, 58syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝑌 = 𝐾)
6059fveq2d 6896 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (Fil‘𝑌) = (Fil‘ 𝐾))
6157, 60eleqtrd 2831 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘ 𝐾))
62 eqid 2728 . . . . . . . . . . 11 𝐾 = 𝐾
6362flimfnfcls 23926 . . . . . . . . . 10 (((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘ 𝐾) → ((𝐹𝐴) ∈ (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)) ↔ ∀ ∈ (Fil‘ 𝐾)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus ))))
6461, 63syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐹𝐴) ∈ (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)) ↔ ∀ ∈ (Fil‘ 𝐾)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus ))))
65 flfval 23888 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐾 fLimf 𝑔)‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)))
6632, 54, 34, 65syl3anc 1369 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐾 fLimf 𝑔)‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)))
6766eleq2d 2815 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹) ↔ (𝐹𝐴) ∈ (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔))))
6860raleqdv 3321 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (∀ ∈ (Fil‘𝑌)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus )) ↔ ∀ ∈ (Fil‘ 𝐾)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus ))))
6964, 67, 683bitr4d 311 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹) ↔ ∀ ∈ (Fil‘𝑌)(((𝑌 FilMap 𝐹)‘𝑔) ⊆ → (𝐹𝐴) ∈ (𝐾 fClus ))))
7051, 69sylibrd 259 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹)))
7170expr 456 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑔 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝑔) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹))))
7271com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → (𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹))))
7372ralrimdva 3150 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) → ∀𝑔 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹))))
7473imdistanda 571 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) → (𝐹:𝑋𝑌 ∧ ∀𝑔 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹)))))
75 cnpflf 23899 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑔 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑔) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑔)‘𝐹)))))
7674, 75sylibrd 259 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
7712, 76impbid 211 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3057  wrex 3066  wss 3945   cuni 4904  wf 6539  cfv 6543  (class class class)co 7415  fBascfbas 21261  Topctop 22789  TopOnctopon 22806   CnP ccnp 23123  Filcfil 23743   FilMap cfm 23831   fLim cflim 23832   fLimf cflf 23833   fClus cfcls 23834   fClusf cfcf 23835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-1o 8481  df-er 8719  df-map 8841  df-en 8959  df-fin 8962  df-fi 9429  df-fbas 21270  df-fg 21271  df-top 22790  df-topon 22807  df-cld 22917  df-ntr 22918  df-cls 22919  df-nei 22996  df-cnp 23126  df-fil 23744  df-fm 23836  df-flim 23837  df-flf 23838  df-fcls 23839  df-fcf 23840
This theorem is referenced by:  cnfcf  23940
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