MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncnp2 Unicode version

Theorem cncnp2 17012
Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
cncnp.1  |-  X  = 
U. J
cncnp.2  |-  Y  = 
U. K
Assertion
Ref Expression
cncnp2  |-  ( X  =/=  (/)  ->  ( F  e.  ( J  Cn  K
)  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
Distinct variable groups:    x, F    x, J    x, K    x, X    x, Y

Proof of Theorem cncnp2
StepHypRef Expression
1 cntop1 16972 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cncnp.1 . . . . . 6  |-  X  = 
U. J
32toptopon 16673 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
5 cntop2 16973 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
6 cncnp.2 . . . . . 6  |-  Y  = 
U. K
76toptopon 16673 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
85, 7sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  (TopOn `  Y )
)
92, 6cnf 16978 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
104, 8, 9jca31 520 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1110adantl 452 . 2  |-  ( ( X  =/=  (/)  /\  F  e.  ( J  Cn  K
) )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
12 r19.2z 3545 . . 3  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )
13 cnptop1 16974 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  J  e.  Top )
1413, 3sylib 188 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  J  e.  (TopOn `  X )
)
15 cnptop2 16975 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  K  e.  Top )
1615, 7sylib 188 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  K  e.  (TopOn `  Y )
)
172, 6cnpf 16979 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  F : X --> Y )
1814, 16, 17jca31 520 . . . 4  |-  ( F  e.  ( ( J  CnP  K ) `  x )  ->  (
( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y ) )
1918rexlimivw 2665 . . 3  |-  ( E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x )  ->  ( ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y ) )
2012, 19syl 15 . 2  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  ( ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  Y ) )  /\  F : X --> Y ) )
21 cncnp 17011 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) ) )
2221baibd 875 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F : X
--> Y )  ->  ( F  e.  ( J  Cn  K )  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
2311, 20, 22pm5.21nd 868 1  |-  ( X  =/=  (/)  ->  ( F  e.  ( J  Cn  K
)  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   (/)c0 3457   U.cuni 3829   -->wf 5253   ` cfv 5257  (class class class)co 5860   Topctop 16633  TopOnctopon 16634    Cn ccn 16956    CnP ccnp 16957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-map 6776  df-topgen 13346  df-top 16638  df-topon 16641  df-cn 16959  df-cnp 16960
  Copyright terms: Public domain W3C validator