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Theorem cncnpi 20371
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 2471 . . . 4  |-  U. K  =  U. K
31, 2cnf 20339 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
43adantr 472 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  F : X --> U. K
)
5 cnima 20358 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  y  e.  K )  ->  ( `' F "
y )  e.  J
)
65ad2ant2r 761 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  -> 
( `' F "
y )  e.  J
)
7 simpr 468 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  A  e.  X )
87adantr 472 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  A  e.  X )
9 simprr 774 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  -> 
( F `  A
)  e.  y )
103ad2antrr 740 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  F : X --> U. K
)
11 ffn 5739 . . . . . . 7  |-  ( F : X --> U. K  ->  F  Fn  X )
12 elpreima 6017 . . . . . . 7  |-  ( F  Fn  X  ->  ( A  e.  ( `' F " y )  <->  ( A  e.  X  /\  ( F `  A )  e.  y ) ) )
1310, 11, 123syl 18 . . . . . 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 936 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  A  e.  ( `' F " y ) )
15 eqimss 3470 . . . . . . . 8  |-  ( x  =  ( `' F " y )  ->  x  C_  ( `' F "
y ) )
1615biantrud 515 . . . . . . 7  |-  ( x  =  ( `' F " y )  ->  ( A  e.  x  <->  ( A  e.  x  /\  x  C_  ( `' F "
y ) ) ) )
17 eleq2 2538 . . . . . . 7  |-  ( x  =  ( `' F " y )  ->  ( A  e.  x  <->  A  e.  ( `' F " y ) ) )
1816, 17bitr3d 263 . . . . . 6  |-  ( x  =  ( `' F " y )  ->  (
( A  e.  x  /\  x  C_  ( `' F " y ) )  <->  A  e.  ( `' F " y ) ) )
1918rspcev 3136 . . . . 5  |-  ( ( ( `' F "
y )  e.  J  /\  A  e.  ( `' F " y ) )  ->  E. x  e.  J  ( A  e.  x  /\  x  C_  ( `' F "
y ) ) )
206, 14, 19syl2anc 673 . . . 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 626 . . 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 2809 . 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 20333 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2423adantr 472 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  J  e.  Top )
251toptopon 20025 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
2624, 25sylib 201 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  J  e.  (TopOn `  X ) )
27 cntop2 20334 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2827adantr 472 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  K  e.  Top )
292toptopon 20025 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
3028, 29sylib 201 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  K  e.  (TopOn `  U. K ) )
31 iscnp3 20337 . . 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 1292 . 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 936 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    C_ wss 3390   U.cuni 4190   `'ccnv 4838   "cima 4842    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995    Cn ccn 20317    CnP ccnp 20318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-top 19998  df-topon 20000  df-cn 20320  df-cnp 20321
This theorem is referenced by:  cnsscnp  20372  cncnp  20373  lmcn  20398  ptcn  20719  tmdcn2  21182  ghmcnp  21207  tsmsmhm  21238  tsmsadd  21239  dvcnp2  22953  dvaddbr  22971  dvmulbr  22972  dvcobr  22979  dvcjbr  22982  dvcnvlem  23007  lhop1lem  23044  dvcnvrelem2  23049  ftc1cn  23074  taylthlem2  23408  psercn  23460  abelth  23475  cxpcn3  23767  efrlim  23974  blocni  26527  cvmlift2lem11  30108  cvmlift2lem12  30109  cvmlift3lem7  30120  poimir  32037  ftc1cnnc  32080  cncfiooicclem1  37868  fouriercn  38208
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