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Theorem cnpfcf 23536
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 22745 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1118 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1168 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 topontop 22406 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
5 cnpfcfi 23535 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))
653com23 1126 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))
763expia 1121 . . . . . . 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 580 . . . . . 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 3150 . . . . 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 1186 . . . 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 512 . . 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 413 . 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 775 . . . . . . . . . . . . . 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 23343 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 774 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 23453 . . . . . . . . . . . 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 23521 . . . . . . . . . . . . . . . . . 18 (𝐽 fLim 𝑓) βŠ† (𝐽 fClus 𝑓)
22 simpll1 1212 . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . . . 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 23467 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑓 ∈ (Filβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fLim 𝑔) βŠ† (𝐽 fLim 𝑓))
2723, 24, 25, 26syl3anc 1371 . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . . . . 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 3979 . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . 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 23528 . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . . . . 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 7421 . . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . 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 692 . . . . . . . . . . 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 457 . . . . . . . . . 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 3154 . . . . . . . 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 22419 . . . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 23439 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 22407 . . . . . . . . . . . . 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 6892 . . . . . . . . . . 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 2835 . . . . . . . . . 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 2732 . . . . . . . . . . 11 βˆͺ 𝐾 = βˆͺ 𝐾
6362flimfnfcls 23523 . . . . . . . . . 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 23485 . . . . . . . . . . 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 1371 . . . . . . . . . 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 2819 . . . . . . . . 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 3325 . . . . . . . . 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 310 . . . . . . . 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 258 . . . . . . 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 457 . . . . . 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 3154 . . . 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 572 . . 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 23496 . . 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 258 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3947  βˆͺ cuni 4907  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  fBascfbas 20924  Topctop 22386  TopOnctopon 22403   CnP ccnp 22720  Filcfil 23340   FilMap cfm 23428   fLim cflim 23429   fLimf cflf 23430   fClus cfcls 23431   fClusf cfcf 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-fin 8939  df-fi 9402  df-fbas 20933  df-fg 20934  df-top 22387  df-topon 22404  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-cnp 22723  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-fcls 23436  df-fcf 23437
This theorem is referenced by:  cnfcf  23537
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