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Theorem cnpfcf 23865
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 23074 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1117 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1167 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 topontop 22735 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
5 cnpfcfi 23864 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))
653com23 1125 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ∧ 𝐴 ∈ (𝐽 fClus 𝑓)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))
763expia 1120 . . . . . . 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 579 . . . . . 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 3149 . . . . 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 1185 . . . 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 511 . . 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 412 . 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 774 . . . . . . . . . . . . . 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 23672 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 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†’ ((π‘Œ FilMap 𝐹)β€˜π‘”) βŠ† β„Ž)
1915, 16, 17, 18fmfnfm 23782 . . . . . . . . . . . 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 3113 . . . . . . . . . . . . 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 23850 . . . . . . . . . . . . . . . . . 18 (𝐽 fLim 𝑓) βŠ† (𝐽 fClus 𝑓)
22 simpll1 1211 . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . 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 23796 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑓 ∈ (Filβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fLim 𝑔) βŠ† (𝐽 fLim 𝑓))
2723, 24, 25, 26syl3anc 1370 . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . 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 3980 . . . . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . 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 23857 . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . . . . 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 7428 . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 3154 . . . . . . . . . . . . 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 691 . . . . . . . . . . 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 456 . . . . . . . . . 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 3153 . . . . . . . 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 22748 . . . . . . . . . . . . 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 768 . . . . . . . . . . . . 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 23768 . . . . . . . . . . . 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 1370 . . . . . . . . . . 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 22736 . . . . . . . . . . . . 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 6895 . . . . . . . . . . 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 2834 . . . . . . . . . 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 2731 . . . . . . . . . . 11 βˆͺ 𝐾 = βˆͺ 𝐾
6362flimfnfcls 23852 . . . . . . . . . 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 23814 . . . . . . . . . . 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 1370 . . . . . . . . . 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 2818 . . . . . . . . 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 3324 . . . . . . . . 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 456 . . . . . 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 3153 . . . 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 571 . . 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 23825 . . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069   βŠ† wss 3948  βˆͺ cuni 4908  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  fBascfbas 21221  Topctop 22715  TopOnctopon 22732   CnP ccnp 23049  Filcfil 23669   FilMap cfm 23757   fLim cflim 23758   fLimf cflf 23759   fClus cfcls 23760   fClusf cfcf 23761
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-1o 8472  df-er 8709  df-map 8828  df-en 8946  df-fin 8949  df-fi 9412  df-fbas 21230  df-fg 21231  df-top 22716  df-topon 22733  df-cld 22843  df-ntr 22844  df-cls 22845  df-nei 22922  df-cnp 23052  df-fil 23670  df-fm 23762  df-flim 23763  df-flf 23764  df-fcls 23765  df-fcf 23766
This theorem is referenced by:  cnfcf  23866
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