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Theorem cnpdis 20251
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 768 . . . . . . . 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 1046 . . . . . . . . 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 3967 . . . . . . . . 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 764 . . . . . . . . . 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 769 . . . . . . . . . . 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 5689 . . . . . . . . . . 11  |-  ( f : X --> Y  -> 
f  Fn  X )
8 elpreima 5961 . . . . . . . . . . 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 930 . . . . . . . . 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 4088 . . . . . . . 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 2495 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( A  e.  y  <-> 
A  e.  { A } ) )
13 sseq1 3428 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( y  C_  ( `' f " x
)  <->  { A }  C_  ( `' f " x
) ) )
1412, 13anbi12d 715 . . . . . . . . 9  |-  ( y  =  { A }  ->  ( ( A  e.  y  /\  y  C_  ( `' f " x
) )  <->  ( A  e.  { A }  /\  { A }  C_  ( `' f " x
) ) ) )
1514rspcev 3125 . . . . . . . 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 1262 . . . . . . 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 618 . . . . . 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 2779 . . . . 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 618 . . . 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 638 . . 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 1009 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  K  e.  (TopOn `  Y ) )
22 toponmax 19885 . . . . 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 1008 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  J  e.  (TopOn `  X ) )
25 toponmax 19885 . . . . 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 7441 . . 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 20202 . . . 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 466 . . 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 289 . 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 2426 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2714   E.wrex 2715    C_ wss 3379   {csn 3941   `'ccnv 4795   "cima 4799    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249    ^m cmap 7427  TopOnctopon 19860    CnP ccnp 20183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-map 7429  df-top 19863  df-topon 19865  df-cnp 20186
This theorem is referenced by: (None)
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