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Theorem cndis 17356
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 5225 . . . . . . . 8  |-  ( `' f " x ) 
C_  dom  f
2 fdm 5596 . . . . . . . . 9  |-  ( f : A --> X  ->  dom  f  =  A
)
32adantl 454 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  dom  f  =  A )
41, 3syl5sseq 3397 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  C_  A )
5 elpw2g 4364 . . . . . . . 8  |-  ( A  e.  V  ->  (
( `' f "
x )  e.  ~P A 
<->  ( `' f "
x )  C_  A
) )
65ad2antrr 708 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( ( `' f " x )  e.  ~P A  <->  ( `' f " x )  C_  A ) )
74, 6mpbird 225 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  e. 
~P A )
87ralrimivw 2791 . . . . 5  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
)
98ex 425 . . . 4  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f : A --> X  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
) )
109pm4.71d 617 . . 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 16994 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
12 id 21 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
13 elmapg 7032 . . . 4  |-  ( ( X  e.  J  /\  A  e.  V )  ->  ( f  e.  ( X  ^m  A )  <-> 
f : A --> X ) )
1411, 12, 13syl2anr 466 . . 3  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( X  ^m  A )  <->  f : A
--> X ) )
15 distopon 17062 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
16 iscn 17300 . . . 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 459 . . 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 279 . 2  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( ~P A  Cn  J )  <-> 
f  e.  ( X  ^m  A ) ) )
1918eqrdv 2435 1  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  ( ~P A  Cn  J
)  =  ( X  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706    C_ wss 3321   ~Pcpw 3800   `'ccnv 4878   dom cdm 4879   "cima 4882   -->wf 5451   ` cfv 5455  (class class class)co 6082    ^m cmap 7019  TopOnctopon 16960    Cn ccn 17289
This theorem is referenced by:  xkopt  17688  distgp  18130  symgtgp  18132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-top 16964  df-topon 16967  df-cn 17292
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