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Theorem cndis 19013
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 5289 . . . . . . . 8  |-  ( `' f " x ) 
C_  dom  f
2 fdm 5663 . . . . . . . . 9  |-  ( f : A --> X  ->  dom  f  =  A
)
32adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  dom  f  =  A )
41, 3syl5sseq 3504 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  C_  A )
5 elpw2g 4555 . . . . . . . 8  |-  ( A  e.  V  ->  (
( `' f "
x )  e.  ~P A 
<->  ( `' f "
x )  C_  A
) )
65ad2antrr 725 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( ( `' f " x )  e.  ~P A  <->  ( `' f " x )  C_  A ) )
74, 6mpbird 232 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  e. 
~P A )
87ralrimivw 2823 . . . . 5  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
)
98ex 434 . . . 4  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f : A --> X  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
) )
109pm4.71d 634 . . 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 18651 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
12 id 22 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
13 elmapg 7329 . . . 4  |-  ( ( X  e.  J  /\  A  e.  V )  ->  ( f  e.  ( X  ^m  A )  <-> 
f : A --> X ) )
1411, 12, 13syl2anr 478 . . 3  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( X  ^m  A )  <->  f : A
--> X ) )
15 distopon 18719 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
16 iscn 18957 . . . 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 471 . . 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 286 . 2  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( ~P A  Cn  J )  <-> 
f  e.  ( X  ^m  A ) ) )
1918eqrdv 2448 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 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3428   ~Pcpw 3960   `'ccnv 4939   dom cdm 4940   "cima 4943   -->wf 5514   ` cfv 5518  (class class class)co 6192    ^m cmap 7316  TopOnctopon 18617    Cn ccn 18946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-top 18621  df-topon 18624  df-cn 18949
This theorem is referenced by:  xkopt  19346  distgp  19788  symgtgp  19790
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