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Theorem cnpdis 12306
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 507 . . . . . . . 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 1005 . . . . . . . . 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 3522 . . . . . . . . 9  |-  ( A  e.  X  ->  A  e.  { A } )
42, 3syl 14 . . . . . . . 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 504 . . . . . . . . . 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 508 . . . . . . . . . . 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 5240 . . . . . . . . . . 11  |-  ( f : X --> Y  -> 
f  Fn  X )
8 elpreima 5505 . . . . . . . . . . 11  |-  ( f  Fn  X  ->  ( A  e.  ( `' f " x )  <->  ( A  e.  X  /\  (
f `  A )  e.  x ) ) )
96, 7, 83syl 17 . . . . . . . . . 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 911 . . . . . . . . 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 3633 . . . . . . . 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 2179 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( A  e.  y  <-> 
A  e.  { A } ) )
13 sseq1 3088 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( y  C_  ( `' f " x
)  <->  { A }  C_  ( `' f " x
) ) )
1412, 13anbi12d 462 . . . . . . . . 9  |-  ( y  =  { A }  ->  ( ( A  e.  y  /\  y  C_  ( `' f " x
) )  <->  ( A  e.  { A }  /\  { A }  C_  ( `' f " x
) ) ) )
1514rspcev 2761 . . . . . . . 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 1197 . . . . . . 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 370 . . . . . 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 2480 . . . . 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 370 . . . 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 388 . . 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 968 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  K  e.  (TopOn `  Y ) )
22 toponmax 12087 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  Y  e.  K )
2321, 22syl 14 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  Y  e.  K )
24 simpl1 967 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  J  e.  (TopOn `  X ) )
25 toponmax 12087 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2624, 25syl 14 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  X  e.  J )
2723, 26elmapd 6522 . . 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 12267 . . . 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 272 . . 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 220 . 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 2113 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 set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392    C_ wss 3039   {csn 3495   `'ccnv 4506   "cima 4510    Fn wfn 5086   -->wf 5087   ` cfv 5091  (class class class)co 5740    ^m cmap 6508  TopOnctopon 12072    CnP ccnp 12250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-map 6510  df-top 12060  df-topon 12073  df-cnp 12253
This theorem is referenced by: (None)
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