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Theorem cnpdis 17021
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 736 . . . . . . . 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 996 . . . . . . . . 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 3665 . . . . . . . . 9  |-  ( A  e.  X  ->  A  e.  { A } )
42, 3syl 15 . . . . . . . 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 733 . . . . . . . . . 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 737 . . . . . . . . . . 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 5389 . . . . . . . . . . 11  |-  ( f : X --> Y  -> 
f  Fn  X )
8 elpreima 5645 . . . . . . . . . . 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 888 . . . . . . . . 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 3760 . . . . . . . 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 2344 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( A  e.  y  <-> 
A  e.  { A } ) )
13 sseq1 3199 . . . . . . . . . 10  |-  ( y  =  { A }  ->  ( y  C_  ( `' f " x
)  <->  { A }  C_  ( `' f " x
) ) )
1412, 13anbi12d 691 . . . . . . . . 9  |-  ( y  =  { A }  ->  ( ( A  e.  y  /\  y  C_  ( `' f " x
) )  <->  ( A  e.  { A }  /\  { A }  C_  ( `' f " x
) ) ) )
1514rspcev 2884 . . . . . . . 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 1180 . . . . . . 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 598 . . . . . 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 2626 . . . . 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 598 . . . 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 615 . . 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 959 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  K  e.  (TopOn `  Y ) )
22 toponmax 16666 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  Y  e.  K )
2321, 22syl 15 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  Y  e.  K )
24 simpl1 958 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  J  e.  (TopOn `  X ) )
25 toponmax 16666 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2624, 25syl 15 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  X  e.  J )
27 elmapg 6785 . . . 4  |-  ( ( Y  e.  K  /\  X  e.  J )  ->  ( f  e.  ( Y  ^m  X )  <-> 
f : X --> Y ) )
2823, 26, 27syl2anc 642 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  { A }  e.  J )  ->  ( f  e.  ( Y  ^m  X )  <-> 
f : X --> Y ) )
29 iscnp3 16974 . . . 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 ) ) ) ) ) )
3029adantr 451 . . 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 ) ) ) ) ) )
3120, 28, 303bitr4rd 277 . 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 ) ) )
3231eqrdv 2281 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   {csn 3640   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772  TopOnctopon 16632    CnP ccnp 16955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cnp 16958
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