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

Theorem cncnpi 19756
Description: A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnsscnp.1  |-  X  = 
U. J
Assertion
Ref Expression
cncnpi  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  F  e.  ( ( J  CnP  K ) `
 A ) )

Proof of Theorem cncnpi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnsscnp.1 . . . 4  |-  X  = 
U. J
2 eqid 2443 . . . 4  |-  U. K  =  U. K
31, 2cnf 19724 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
43adantr 465 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  F : X --> U. K
)
5 cnima 19743 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  y  e.  K )  ->  ( `' F "
y )  e.  J
)
65ad2ant2r 746 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  -> 
( `' F "
y )  e.  J
)
7 simpr 461 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  A  e.  X )
87adantr 465 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  A  e.  X )
9 simprr 757 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  -> 
( F `  A
)  e.  y )
103ad2antrr 725 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  F : X --> U. K
)
11 ffn 5721 . . . . . . 7  |-  ( F : X --> U. K  ->  F  Fn  X )
12 elpreima 5992 . . . . . . 7  |-  ( F  Fn  X  ->  ( A  e.  ( `' F " y )  <->  ( A  e.  X  /\  ( F `  A )  e.  y ) ) )
1310, 11, 123syl 20 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  -> 
( A  e.  ( `' F " y )  <-> 
( A  e.  X  /\  ( F `  A
)  e.  y ) ) )
148, 9, 13mpbir2and 922 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  A  e.  ( `' F " y ) )
15 eqimss 3541 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  x  C_  ( `' F "
y ) )
1615biantrud 507 . . . . . . 7  |-  ( x  =  ( `' F " y )  ->  ( A  e.  x  <->  ( A  e.  x  /\  x  C_  ( `' F "
y ) ) ) )
17 eleq2 2516 . . . . . . 7  |-  ( x  =  ( `' F " y )  ->  ( A  e.  x  <->  A  e.  ( `' F " y ) ) )
1816, 17bitr3d 255 . . . . . 6  |-  ( x  =  ( `' F " y )  ->  (
( A  e.  x  /\  x  C_  ( `' F " y ) )  <->  A  e.  ( `' F " y ) ) )
1918rspcev 3196 . . . . 5  |-  ( ( ( `' F "
y )  e.  J  /\  A  e.  ( `' F " y ) )  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F "
y ) ) )
206, 14, 19syl2anc 661 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F " y ) ) )
2120expr 615 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  y  e.  K )  ->  (
( F `  A
)  e.  y  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F " y ) ) ) )
2221ralrimiva 2857 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  A. y  e.  K  ( ( F `  A )  e.  y  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F " y ) ) ) )
23 cntop1 19718 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2423adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  J  e.  Top )
251toptopon 19411 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
2624, 25sylib 196 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  J  e.  (TopOn `  X ) )
27 cntop2 19719 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2827adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  K  e.  Top )
292toptopon 19411 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
3028, 29sylib 196 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  K  e.  (TopOn `  U. K ) )
31 iscnp3 19722 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  A  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F : X --> U. K  /\  A. y  e.  K  ( ( F `  A )  e.  y  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F "
y ) ) ) ) ) )
3226, 30, 7, 31syl3anc 1229 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  ( F  e.  ( ( J  CnP  K
) `  A )  <->  ( F : X --> U. K  /\  A. y  e.  K  ( ( F `  A )  e.  y  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F " y ) ) ) ) ) )
334, 22, 32mpbir2and 922 1  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  F  e.  ( ( J  CnP  K ) `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   U.cuni 4234   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   Topctop 19371  TopOnctopon 19372    Cn ccn 19702    CnP ccnp 19703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-map 7424  df-top 19376  df-topon 19379  df-cn 19705  df-cnp 19706
This theorem is referenced by:  cnsscnp  19757  cncnp  19758  lmcn  19783  ptcn  20105  tmdcn2  20565  ghmcnp  20590  tsmsmhm  20625  tsmsadd  20626  dvcnp2  22300  dvaddbr  22318  dvmulbr  22319  dvcobr  22326  dvcjbr  22329  dvcnvlem  22354  lhop1lem  22391  dvcnvrelem2  22396  ftc1cn  22421  taylthlem2  22745  psercn  22797  abelth  22812  cxpcn3  23098  efrlim  23275  blocni  25696  cvmlift2lem11  28735  cvmlift2lem12  28736  cvmlift3lem7  28747  ftc1cnnc  30064  cncfiooicclem1  31603  fouriercn  31904
  Copyright terms: Public domain W3C validator