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Theorem cnpdis 20012
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 761 . . . . . . . 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 1037 . . . . . . . . 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 4058 . . . . . . . . 9  |-  ( A  e.  X  ->  A  e.  { A } )
42, 3syl 16 . . . . . . . 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 757 . . . . . . . . . 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 762 . . . . . . . . . . 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 5737 . . . . . . . . . . 11  |-  ( f : X --> Y  -> 
f  Fn  X )
8 elpreima 6008 . . . . . . . . . . 11  |-  ( f  Fn  X  ->  ( A  e.  ( `' f " x )  <->  ( A  e.  X  /\  (
f `  A )  e.  x ) ) )
96, 7, 83syl 20 . . . . . . . . . 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 922 . . . . . . . . 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 4177 . . . . . . . 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 2530 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( A  e.  y  <-> 
A  e.  { A } ) )
13 sseq1 3520 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( y  C_  ( `' f " x
)  <->  { A }  C_  ( `' f " x
) ) )
1412, 13anbi12d 710 . . . . . . . . 9  |-  ( y  =  { A }  ->  ( ( A  e.  y  /\  y  C_  ( `' f " x
) )  <->  ( A  e.  { A }  /\  { A }  C_  ( `' f " x
) ) ) )
1514rspcev 3210 . . . . . . . 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 1226 . . . . . . 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 615 . . . . . 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 2871 . . . . 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 615 . . . 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 634 . . 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 1000 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  K  e.  (TopOn `  Y ) )
22 toponmax 19647 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  Y  e.  K )
2321, 22syl 16 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  Y  e.  K )
24 simpl1 999 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  J  e.  (TopOn `  X ) )
25 toponmax 19647 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2624, 25syl 16 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  X  e.  J )
2723, 26elmapd 7452 . . 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 19963 . . . 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 465 . . 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 286 . 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 2454 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   {csn 4032   `'ccnv 5007   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438  TopOnctopon 19613    CnP ccnp 19944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-top 19617  df-topon 19620  df-cnp 19947
This theorem is referenced by: (None)
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