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Theorem cncnpi 18885
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 18853 . . 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 18872 . . . . . 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 756 . . . . . 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 5562 . . . . . . 7  |-  ( F : X --> U. K  ->  F  Fn  X )
12 elpreima 5826 . . . . . . 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 913 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  e.  X
)  /\  ( y  e.  K  /\  ( F `  A )  e.  y ) )  ->  A  e.  ( `' F " y ) )
15 eqimss 3411 . . . . . . . 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 2504 . . . . . . 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 3076 . . . . 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 2802 . 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 18847 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2423adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  J  e.  Top )
251toptopon 18541 . . . 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 18848 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2827adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  K  e.  Top )
292toptopon 18541 . . . 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 18851 . . 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 1218 . 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 913 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 1369    e. wcel 1756   A.wral 2718   E.wrex 2719    C_ wss 3331   U.cuni 4094   `'ccnv 4842   "cima 4846    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094   Topctop 18501  TopOnctopon 18502    Cn ccn 18831    CnP ccnp 18832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-map 7219  df-top 18506  df-topon 18509  df-cn 18834  df-cnp 18835
This theorem is referenced by:  cnsscnp  18886  cncnp  18887  lmcn  18912  ptcn  19203  tmdcn2  19663  ghmcnp  19688  tsmsmhm  19723  tsmsadd  19724  dvcnp2  21397  dvaddbr  21415  dvmulbr  21416  dvcobr  21423  dvcjbr  21426  dvcnvlem  21451  lhop1lem  21488  dvcnvrelem2  21493  ftc1cn  21518  taylthlem2  21842  psercn  21894  abelth  21909  cxpcn3  22189  efrlim  22366  blocni  24208  cvmlift2lem11  27205  cvmlift2lem12  27206  cvmlift3lem7  27217  ftc1cnnc  28469
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