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Theorem cnpfcf 23408
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐹   𝑓,𝐽   𝑓,𝐾   𝑓,𝑋   𝑓,π‘Œ

Proof of Theorem cnpfcf
Dummy variables 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnpf2 22617 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1119 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1169 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 topontop 22278 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
5 cnpfcfi 23407 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))
653com23 1127 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))
763expia 1122 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
84, 7sylan 581 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
98ralrimivw 3144 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
1093ad2antl2 1187 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
113, 10jca 513 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
1211ex 414 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))))
13 simplrl 776 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
14 filfbas 23215 . . . . . . . . . . . . . 14 (𝑔 ∈ (Filβ€˜π‘‹) β†’ 𝑔 ∈ (fBasβ€˜π‘‹))
1513, 14syl 17 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ 𝑔 ∈ (fBasβ€˜π‘‹))
16 simprl 770 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ β„Ž ∈ (Filβ€˜π‘Œ))
17 simpllr 775 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
18 simprr 772 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)
1915, 16, 17, 18fmfnfm 23325 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ βˆƒπ‘“ ∈ (Filβ€˜π‘‹)(𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))
20 r19.29 3114 . . . . . . . . . . . . 13 ((βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ∧ βˆƒπ‘“ ∈ (Filβ€˜π‘‹)(𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“))) β†’ βˆƒπ‘“ ∈ (Filβ€˜π‘‹)((𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“))))
21 flimfcls 23393 . . . . . . . . . . . . . . . . . 18 (𝐽 fLim 𝑓) βŠ† (𝐽 fClus 𝑓)
22 simpll1 1213 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2322ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
24 simprl 770 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝑓 ∈ (Filβ€˜π‘‹))
25 simprrl 780 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝑔 βŠ† 𝑓)
26 flimss2 23339 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑓 ∈ (Filβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fLim 𝑔) βŠ† (𝐽 fLim 𝑓))
2723, 24, 25, 26syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ (𝐽 fLim 𝑔) βŠ† (𝐽 fLim 𝑓))
28 simprr 772 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ 𝐴 ∈ (𝐽 fLim 𝑔))
2928ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝐴 ∈ (𝐽 fLim 𝑔))
3027, 29sseldd 3946 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝐴 ∈ (𝐽 fLim 𝑓))
3121, 30sselid 3943 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝐴 ∈ (𝐽 fClus 𝑓))
32 simpll2 1214 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3332ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
34 simplr 768 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3534ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
36 fcfval 23400 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑓 ∈ (Filβ€˜π‘‹) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((𝐾 fClusf 𝑓)β€˜πΉ) = (𝐾 fClus ((π‘Œ FilMap 𝐹)β€˜π‘“)))
3733, 24, 35, 36syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ ((𝐾 fClusf 𝑓)β€˜πΉ) = (𝐾 fClus ((π‘Œ FilMap 𝐹)β€˜π‘“)))
38 simprrr 781 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“))
3938oveq2d 7374 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ (𝐾 fClus β„Ž) = (𝐾 fClus ((π‘Œ FilMap 𝐹)β€˜π‘“)))
4037, 39eqtr4d 2776 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ ((𝐾 fClusf 𝑓)β€˜πΉ) = (𝐾 fClus β„Ž))
4140eleq2d 2820 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ) ↔ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4241biimpd 228 . . . . . . . . . . . . . . . . 17 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4331, 42embantd 59 . . . . . . . . . . . . . . . 16 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ (𝑓 ∈ (Filβ€˜π‘‹) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)))) β†’ ((𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4443expr 458 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ 𝑓 ∈ (Filβ€˜π‘‹)) β†’ ((𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“)) β†’ ((𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
4544impcomd 413 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) ∧ 𝑓 ∈ (Filβ€˜π‘‹)) β†’ (((𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“))) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4645rexlimdva 3149 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ (βˆƒπ‘“ ∈ (Filβ€˜π‘‹)((𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ∧ (𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“))) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4720, 46syl5 34 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ ((βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ∧ βˆƒπ‘“ ∈ (Filβ€˜π‘‹)(𝑔 βŠ† 𝑓 ∧ β„Ž = ((π‘Œ FilMap 𝐹)β€˜π‘“))) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4819, 47mpan2d 693 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ (β„Ž ∈ (Filβ€˜π‘Œ) ∧ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)))
4948expr 458 . . . . . . . . . 10 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ β„Ž ∈ (Filβ€˜π‘Œ)) β†’ (((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
5049com23 86 . . . . . . . . 9 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) ∧ β„Ž ∈ (Filβ€˜π‘Œ)) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
5150ralrimdva 3148 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ βˆ€β„Ž ∈ (Filβ€˜π‘Œ)(((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
52 toponmax 22291 . . . . . . . . . . . . 13 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐾)
5332, 52syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ π‘Œ ∈ 𝐾)
54 simprl 770 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
5554, 14syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ 𝑔 ∈ (fBasβ€˜π‘‹))
56 fmfil 23311 . . . . . . . . . . . 12 ((π‘Œ ∈ 𝐾 ∧ 𝑔 ∈ (fBasβ€˜π‘‹) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘Œ FilMap 𝐹)β€˜π‘”) ∈ (Filβ€˜π‘Œ))
5753, 55, 34, 56syl3anc 1372 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ ((π‘Œ FilMap 𝐹)β€˜π‘”) ∈ (Filβ€˜π‘Œ))
58 toponuni 22279 . . . . . . . . . . . . 13 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
5932, 58syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ π‘Œ = βˆͺ 𝐾)
6059fveq2d 6847 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ (Filβ€˜π‘Œ) = (Filβ€˜βˆͺ 𝐾))
6157, 60eleqtrd 2836 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ ((π‘Œ FilMap 𝐹)β€˜π‘”) ∈ (Filβ€˜βˆͺ 𝐾))
62 eqid 2733 . . . . . . . . . . 11 βˆͺ 𝐾 = βˆͺ 𝐾
6362flimfnfcls 23395 . . . . . . . . . 10 (((π‘Œ FilMap 𝐹)β€˜π‘”) ∈ (Filβ€˜βˆͺ 𝐾) β†’ ((πΉβ€˜π΄) ∈ (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜π‘”)) ↔ βˆ€β„Ž ∈ (Filβ€˜βˆͺ 𝐾)(((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
6461, 63syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ ((πΉβ€˜π΄) ∈ (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜π‘”)) ↔ βˆ€β„Ž ∈ (Filβ€˜βˆͺ 𝐾)(((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
65 flfval 23357 . . . . . . . . . . 11 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((𝐾 fLimf 𝑔)β€˜πΉ) = (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜π‘”)))
6632, 54, 34, 65syl3anc 1372 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ ((𝐾 fLimf 𝑔)β€˜πΉ) = (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜π‘”)))
6766eleq2d 2820 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ) ↔ (πΉβ€˜π΄) ∈ (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜π‘”))))
6860raleqdv 3312 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ (βˆ€β„Ž ∈ (Filβ€˜π‘Œ)(((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž)) ↔ βˆ€β„Ž ∈ (Filβ€˜βˆͺ 𝐾)(((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
6964, 67, 683bitr4d 311 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ) ↔ βˆ€β„Ž ∈ (Filβ€˜π‘Œ)(((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž β†’ (πΉβ€˜π΄) ∈ (𝐾 fClus β„Ž))))
7051, 69sylibrd 259 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fLim 𝑔))) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ)))
7170expr 458 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝑔) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ))))
7271com23 86 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ (𝐴 ∈ (𝐽 fLim 𝑔) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ))))
7372ralrimdva 3148 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) β†’ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑔) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ))))
7473imdistanda 573 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑔) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ)))))
75 cnpflf 23368 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑔) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑔)β€˜πΉ)))))
7674, 75sylibrd 259 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)))
7712, 76impbid 211 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3911  βˆͺ cuni 4866  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  fBascfbas 20800  Topctop 22258  TopOnctopon 22275   CnP ccnp 22592  Filcfil 23212   FilMap cfm 23300   fLim cflim 23301   fLimf cflf 23302   fClus cfcls 23303   fClusf cfcf 23304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-fin 8890  df-fi 9352  df-fbas 20809  df-fg 20810  df-top 22259  df-topon 22276  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-cnp 22595  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-fcls 23308  df-fcf 23309
This theorem is referenced by:  cnfcf  23409
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