MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cndis Structured version   Unicode version

Theorem cndis 19660
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 5363 . . . . . . . 8  |-  ( `' f " x ) 
C_  dom  f
2 fdm 5741 . . . . . . . . 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 3557 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  C_  A )
5 elpw2g 4616 . . . . . . . 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 2882 . . . . 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 19298 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
12 id 22 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
13 elmapg 7445 . . . 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 19366 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
16 iscn 19604 . . . 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 2464 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 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   ~Pcpw 4016   `'ccnv 5004   dom cdm 5005   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432  TopOnctopon 19264    Cn ccn 19593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-top 19268  df-topon 19271  df-cn 19596
This theorem is referenced by:  xkopt  20024  distgp  20466  symgtgp  20468
  Copyright terms: Public domain W3C validator