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Theorem cndis 13634
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 4991 . . . . . . . 8  |-  ( `' f " x ) 
C_  dom  f
2 fdm 5371 . . . . . . . . 9  |-  ( f : A --> X  ->  dom  f  =  A
)
32adantl 277 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  dom  f  =  A )
41, 3sseqtrid 3205 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  C_  A )
5 elpw2g 4156 . . . . . . . 8  |-  ( A  e.  V  ->  (
( `' f "
x )  e.  ~P A 
<->  ( `' f "
x )  C_  A
) )
65ad2antrr 488 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( ( `' f " x )  e.  ~P A  <->  ( `' f " x )  C_  A ) )
74, 6mpbird 167 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  ( `' f
" x )  e. 
~P A )
87ralrimivw 2551 . . . . 5  |-  ( ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  /\  f : A --> X )  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
)
98ex 115 . . . 4  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f : A --> X  ->  A. x  e.  J  ( `' f " x
)  e.  ~P A
) )
109pm4.71d 393 . . 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 13416 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
12 id 19 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
13 elmapg 6660 . . . 4  |-  ( ( X  e.  J  /\  A  e.  V )  ->  ( f  e.  ( X  ^m  A )  <-> 
f : A --> X ) )
1411, 12, 13syl2anr 290 . . 3  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( X  ^m  A )  <->  f : A
--> X ) )
15 distopon 13480 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
16 iscn 13590 . . . 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 283 . . 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 221 . 2  |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X
) )  ->  (
f  e.  ( ~P A  Cn  J )  <-> 
f  e.  ( X  ^m  A ) ) )
1918eqrdv 2175 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    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   ~Pcpw 3575   `'ccnv 4625   dom cdm 4626   "cima 4629   -->wf 5212   ` cfv 5216  (class class class)co 5874    ^m cmap 6647  TopOnctopon 13401    Cn ccn 13578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-map 6649  df-top 13389  df-topon 13402  df-cn 13581
This theorem is referenced by: (None)
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