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Theorem cnpfcf 24049
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 23258 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
213expa 1119 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
323adantl3 1169 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
4 topontop 22919 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5 cnpfcfi 24048 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
653com23 1127 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹))
763expia 1122 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
84, 7sylan 580 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
98ralrimivw 3150 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))
1093ad2antl2 1187 . . . 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 777 . . . . . . . . . . . . . 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 23856 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 23966 . . . . . . . . . . . 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 3114 . . . . . . . . . . . . 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 24034 . . . . . . . . . . . . . . . . . 18 (𝐽 fLim 𝑓) ⊆ (𝐽 fClus 𝑓)
22 simpll1 1213 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐽 ∈ (TopOn‘𝑋))
2322ad2antrr 726 . . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . . 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 23980 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑔𝑓) → (𝐽 fLim 𝑔) ⊆ (𝐽 fLim 𝑓))
2723, 24, 25, 26syl3anc 1373 . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐴 ∈ (𝐽 fLim 𝑔))
2928ad2antrr 726 . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . 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 3981 . . . . . . . . . . . . . . . . 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 1214 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐾 ∈ (TopOn‘𝑌))
3332ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → 𝐹:𝑋𝑌)
3534ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 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 24041 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((𝑌 FilMap 𝐹)‘𝑓)))
3733, 24, 35, 36syl3anc 1373 . . . . . . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . 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 694 . . . . . . . . . . 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 3154 . . . . . . . 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 22932 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 23952 . . . . . . . . . . . 12 ((𝑌𝐾𝑔 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑌 FilMap 𝐹)‘𝑔) ∈ (Fil‘𝑌))
5753, 55, 34, 56syl3anc 1373 . . . . . . . . . . 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 22920 . . . . . . . . . . . . 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 6910 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) → (Fil‘𝑌) = (Fil‘ 𝐾))
6157, 60eleqtrd 2843 . . . . . . . . . 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 2737 . . . . . . . . . . 11 𝐾 = 𝐾
6362flimfnfcls 24036 . . . . . . . . . 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 23998 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐾 fLimf 𝑔)‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘𝑔)))
6632, 54, 34, 65syl3anc 1373 . . . . . . . . . 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 2827 . . . . . . . . 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 3326 . . . . . . . . 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 3154 . . . 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 24009 . . 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 212 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   cuni 4907  wf 6557  cfv 6561  (class class class)co 7431  fBascfbas 21352  Topctop 22899  TopOnctopon 22916   CnP ccnp 23233  Filcfil 23853   FilMap cfm 23941   fLim cflim 23942   fLimf cflf 23943   fClus cfcls 23944   fClusf cfcf 23945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986  df-fin 8989  df-fi 9451  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cnp 23236  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-fcls 23949  df-fcf 23950
This theorem is referenced by:  cnfcf  24050
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