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Theorem cnpdis 20357
Description: If  A is an isolated point in  X (or equivalently, the singleton  { A } is open in  X), then every function is continuous at  A. (Contributed by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
cnpdis  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( ( J  CnP  K ) `  A )  =  ( Y  ^m  X ) )

Proof of Theorem cnpdis
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplrl 775 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  ->  { A }  e.  J
)
2 simpll3 1055 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  ->  A  e.  X )
3 snidg 4005 . . . . . . . . 9  |-  ( A  e.  X  ->  A  e.  { A } )
42, 3syl 17 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  ->  A  e.  { A } )
5 simprr 771 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  -> 
( f `  A
)  e.  x )
6 simplrr 776 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  -> 
f : X --> Y )
7 ffn 5750 . . . . . . . . . . 11  |-  ( f : X --> Y  -> 
f  Fn  X )
8 elpreima 6024 . . . . . . . . . . 11  |-  ( f  Fn  X  ->  ( A  e.  ( `' f " x )  <->  ( A  e.  X  /\  (
f `  A )  e.  x ) ) )
96, 7, 83syl 18 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  -> 
( A  e.  ( `' f " x
)  <->  ( A  e.  X  /\  ( f `
 A )  e.  x ) ) )
102, 5, 9mpbir2and 938 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  ->  A  e.  ( `' f " x ) )
1110snssd 4129 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  ->  { A }  C_  ( `' f " x
) )
12 eleq2 2528 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( A  e.  y  <-> 
A  e.  { A } ) )
13 sseq1 3464 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( y  C_  ( `' f " x
)  <->  { A }  C_  ( `' f " x
) ) )
1412, 13anbi12d 722 . . . . . . . . 9  |-  ( y  =  { A }  ->  ( ( A  e.  y  /\  y  C_  ( `' f " x
) )  <->  ( A  e.  { A }  /\  { A }  C_  ( `' f " x
) ) ) )
1514rspcev 3161 . . . . . . . 8  |-  ( ( { A }  e.  J  /\  ( A  e. 
{ A }  /\  { A }  C_  ( `' f " x
) ) )  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f " x ) ) )
161, 4, 11, 15syl12anc 1274 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  ( x  e.  K  /\  (
f `  A )  e.  x ) )  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f " x ) ) )
1716expr 624 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  ( { A }  e.  J  /\  f : X --> Y ) )  /\  x  e.  K )  ->  (
( f `  A
)  e.  x  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f " x ) ) ) )
1817ralrimiva 2813 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  ( { A }  e.  J  /\  f : X --> Y ) )  ->  A. x  e.  K  ( (
f `  A )  e.  x  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f "
x ) ) ) )
1918expr 624 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( f : X --> Y  ->  A. x  e.  K  ( ( f `  A )  e.  x  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f " x ) ) ) ) )
2019pm4.71d 644 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( f : X --> Y 
<->  ( f : X --> Y  /\  A. x  e.  K  ( ( f `
 A )  e.  x  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f "
x ) ) ) ) ) )
21 simpl2 1018 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  K  e.  (TopOn `  Y ) )
22 toponmax 19991 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  Y  e.  K )
2321, 22syl 17 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  Y  e.  K )
24 simpl1 1017 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  J  e.  (TopOn `  X ) )
25 toponmax 19991 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2624, 25syl 17 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  X  e.  J )
2723, 26elmapd 7511 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( f  e.  ( Y  ^m  X )  <-> 
f : X --> Y ) )
28 iscnp3 20308 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( f  e.  ( ( J  CnP  K ) `  A )  <-> 
( f : X --> Y  /\  A. x  e.  K  ( ( f `
 A )  e.  x  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f "
x ) ) ) ) ) )
2928adantr 471 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( f  e.  ( ( J  CnP  K
) `  A )  <->  ( f : X --> Y  /\  A. x  e.  K  ( ( f `  A
)  e.  x  ->  E. y  e.  J  ( A  e.  y  /\  y  C_  ( `' f " x ) ) ) ) ) )
3020, 27, 293bitr4rd 294 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( f  e.  ( ( J  CnP  K
) `  A )  <->  f  e.  ( Y  ^m  X ) ) )
3130eqrdv 2459 1  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( ( J  CnP  K ) `  A )  =  ( Y  ^m  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   A.wral 2748   E.wrex 2749    C_ wss 3415   {csn 3979   `'ccnv 4851   "cima 4855    Fn wfn 5595   -->wf 5596   ` cfv 5600  (class class class)co 6314    ^m cmap 7497  TopOnctopon 19966    CnP ccnp 20289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-map 7499  df-top 19969  df-topon 19971  df-cnp 20292
This theorem is referenced by: (None)
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