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Theorem cnnei 19542
Description: Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
Hypotheses
Ref Expression
cnnei.x  |-  X  = 
U. J
cnnei.y  |-  Y  = 
U. K
Assertion
Ref Expression
cnnei  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F : X --> Y )  -> 
( F  e.  ( J  Cn  K )  <->  A. p  e.  X  A. w  e.  (
( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
Distinct variable groups:    v, p, w, F    J, p, v, w    K, p, v, w    X, p, v, w    Y, p, v, w

Proof of Theorem cnnei
StepHypRef Expression
1 cnnei.x . . . . . 6  |-  X  = 
U. J
21toptopon 19194 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 cnnei.y . . . . . 6  |-  Y  = 
U. K
43toptopon 19194 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
52, 4anbi12i 697 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top )  <->  ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
) )
6 cncnp 19540 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. p  e.  X  F  e.  ( ( J  CnP  K ) `  p ) ) ) )
76baibd 902 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( F  e.  ( J  Cn  K )  <->  A. p  e.  X  F  e.  ( ( J  CnP  K ) `  p ) ) )
85, 7sylanb 472 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  F : X --> Y )  ->  ( F  e.  ( J  Cn  K
)  <->  A. p  e.  X  F  e.  ( ( J  CnP  K ) `  p ) ) )
95anbi1i 695 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  F : X --> Y )  <-> 
( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y ) )
10 iscnp4 19523 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  p  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  p )  <-> 
( F : X --> Y  /\  A. w  e.  ( ( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) ) )
11103expa 1191 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  p  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  p )  <->  ( F : X --> Y  /\  A. w  e.  ( ( nei `  K ) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) ) )
1211baibd 902 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  p  e.  X )  /\  F : X --> Y )  ->  ( F  e.  ( ( J  CnP  K ) `  p )  <->  A. w  e.  (
( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
1312an32s 802 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y )  /\  p  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  p )  <->  A. w  e.  (
( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
149, 13sylanb 472 . . . 4  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  F : X
--> Y )  /\  p  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  p )  <->  A. w  e.  ( ( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
1514ralbidva 2893 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  F : X --> Y )  ->  ( A. p  e.  X  F  e.  ( ( J  CnP  K ) `  p )  <->  A. p  e.  X  A. w  e.  (
( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
168, 15bitrd 253 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  F : X --> Y )  ->  ( F  e.  ( J  Cn  K
)  <->  A. p  e.  X  A. w  e.  (
( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
17163impa 1186 1  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F : X --> Y )  -> 
( F  e.  ( J  Cn  K )  <->  A. p  e.  X  A. w  e.  (
( nei `  K
) `  { ( F `  p ) } ) E. v  e.  ( ( nei `  J
) `  { p } ) ( F
" v )  C_  w ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   {csn 4020   U.cuni 4238   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275   Topctop 19154  TopOnctopon 19155   neicnei 19357    Cn ccn 19484    CnP ccnp 19485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-topgen 14688  df-top 19159  df-topon 19162  df-ntr 19280  df-nei 19358  df-cn 19487  df-cnp 19488
This theorem is referenced by:  cnextcn  20295  cnextfres  20296
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