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Theorem cndis 20355
Description: Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cndis  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  ( ~P A  Cn  J
)  =  ( X  ^m  A ) )

Proof of Theorem cndis
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 5206 . . . . . . . 8  |-  ( `' f " x ) 
C_  dom  f
2 fdm 5755 . . . . . . . . 9  |-  ( f : A --> X  ->  dom  f  =  A
)
32adantl 472 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  dom  f  =  A )
41, 3syl5sseq 3491 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  C_  A )
5 elpw2g 4579 . . . . . . . 8  |-  ( A  e.  V  ->  (
( `' f "
x )  e.  ~P A 
<->  ( `' f "
x )  C_  A
) )
65ad2antrr 737 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( ( `' f " x )  e.  ~P A  <->  ( `' f " x )  C_  A ) )
74, 6mpbird 240 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  e. 
~P A )
87ralrimivw 2814 . . . . 5  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
)
98ex 440 . . . 4  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f : A --> X  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
) )
109pm4.71d 644 . . 3  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f : A --> X  <->  ( f : A --> X  /\  A. x  e.  J  ( `' f " x
)  e.  ~P A
) ) )
11 toponmax 19991 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
12 id 22 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
13 elmapg 7510 . . . 4  |-  ( ( X  e.  J  /\  A  e.  V )  ->  ( f  e.  ( X  ^m  A )  <-> 
f : A --> X ) )
1411, 12, 13syl2anr 485 . . 3  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( X  ^m  A )  <->  f : A
--> X ) )
15 distopon 20060 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
16 iscn 20299 . . . 4  |-  ( ( ~P A  e.  (TopOn `  A )  /\  J  e.  (TopOn `  X )
)  ->  ( f  e.  ( ~P A  Cn  J )  <->  ( f : A --> X  /\  A. x  e.  J  ( `' f " x
)  e.  ~P A
) ) )
1715, 16sylan 478 . . 3  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( ~P A  Cn  J )  <-> 
( f : A --> X  /\  A. x  e.  J  ( `' f
" x )  e. 
~P A ) ) )
1810, 14, 173bitr4rd 294 . 2  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( ~P A  Cn  J )  <-> 
f  e.  ( X  ^m  A ) ) )
1918eqrdv 2459 1  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  ( ~P A  Cn  J
)  =  ( X  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748    C_ wss 3415   ~Pcpw 3962   `'ccnv 4851   dom cdm 4852   "cima 4855   -->wf 5596   ` cfv 5600  (class class class)co 6314    ^m cmap 7497  TopOnctopon 19966    Cn ccn 20288
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-cn 20291
This theorem is referenced by:  xkopt  20718  distgp  21162  symgtgp  21164
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