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Theorem cnindis 20308
Description: Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnindis  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  ( J  Cn  { (/) ,  A } )  =  ( A  ^m  X ) )

Proof of Theorem cnindis
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpri 3985 . . . . . . 7  |-  ( x  e.  { (/) ,  A }  ->  ( x  =  (/)  \/  x  =  A ) )
2 topontop 19941 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
32ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  J  e.  Top )
4 0opn 19934 . . . . . . . . . 10  |-  ( J  e.  Top  ->  (/)  e.  J
)
53, 4syl 17 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  (/) 
e.  J )
6 imaeq2 5164 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( `' f " x )  =  ( `' f
" (/) ) )
7 ima0 5183 . . . . . . . . . . 11  |-  ( `' f " (/) )  =  (/)
86, 7syl6eq 2501 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( `' f " x )  =  (/) )
98eleq1d 2513 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ( `' f " x
)  e.  J  <->  (/)  e.  J
) )
105, 9syl5ibrcom 226 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  (/)  ->  ( `' f "
x )  e.  J
) )
11 fimacnv 6012 . . . . . . . . . . 11  |-  ( f : X --> A  -> 
( `' f " A )  =  X )
1211adantl 468 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  =  X )
13 toponmax 19943 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
1413ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  X  e.  J )
1512, 14eqeltrd 2529 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  e.  J
)
16 imaeq2 5164 . . . . . . . . . 10  |-  ( x  =  A  ->  ( `' f " x
)  =  ( `' f " A ) )
1716eleq1d 2513 . . . . . . . . 9  |-  ( x  =  A  ->  (
( `' f "
x )  e.  J  <->  ( `' f " A
)  e.  J ) )
1815, 17syl5ibrcom 226 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  A  ->  ( `' f
" x )  e.  J ) )
1910, 18jaod 382 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( ( x  =  (/)  \/  x  =  A )  ->  ( `' f " x )  e.  J ) )
201, 19syl5 33 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  e.  { (/)
,  A }  ->  ( `' f " x
)  e.  J ) )
2120ralrimiv 2800 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J )
2221ex 436 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J ) )
2322pm4.71d 640 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  <->  ( f : X --> A  /\  A. x  e.  { (/) ,  A }  ( `' f
" x )  e.  J ) ) )
24 id 22 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
25 elmapg 7485 . . . 4  |-  ( ( A  e.  V  /\  X  e.  J )  ->  ( f  e.  ( A  ^m  X )  <-> 
f : X --> A ) )
2624, 13, 25syl2anr 481 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( A  ^m  X )  <->  f : X
--> A ) )
27 indistopon 20016 . . . 4  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
28 iscn 20251 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { (/)
,  A }  e.  (TopOn `  A ) )  ->  ( f  e.  ( J  Cn  { (/)
,  A } )  <-> 
( f : X --> A  /\  A. x  e. 
{ (/) ,  A } 
( `' f "
x )  e.  J
) ) )
2927, 28sylan2 477 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( J  Cn  { (/) ,  A } )  <->  ( f : X --> A  /\  A. x  e.  { (/) ,  A }  ( `' f
" x )  e.  J ) ) )
3023, 26, 293bitr4rd 290 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( J  Cn  { (/) ,  A } )  <->  f  e.  ( A  ^m  X ) ) )
3130eqrdv 2449 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  ( J  Cn  { (/) ,  A } )  =  ( A  ^m  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   (/)c0 3731   {cpr 3970   `'ccnv 4833   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290    ^m cmap 7472   Topctop 19917  TopOnctopon 19918    Cn ccn 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-map 7474  df-top 19921  df-topon 19923  df-cn 20243
This theorem is referenced by:  indishmph  20813  indistgp  21115  indispcon  29957
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