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

Theorem cncnp2 17006
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 16966 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 cncnp.1 . . . . . 6  |-  X  = 
U. J
32toptopon 16667 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
41, 3sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
5 cntop2 16967 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
6 cncnp.2 . . . . . 6  |-  Y  = 
U. K
76toptopon 16667 . . . . 5  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
85, 7sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  (TopOn `  Y )
)
92, 6cnf 16972 . . . 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 3544 . . 3  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )  ->  E. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) )
13 cnptop1 16968 . . . . . 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 16969 . . . . . 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 16973 . . . . 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 2664 . . 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 17005 . . 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 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   (/)c0 3456   U.cuni 3828   -->wf 5217   ` cfv 5221  (class class class)co 5820   Topctop 16627  TopOnctopon 16628    Cn ccn 16950    CnP ccnp 16951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-map 6770  df-topgen 13340  df-top 16632  df-topon 16635  df-cn 16953  df-cnp 16954
  Copyright terms: Public domain W3C validator