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Theorem cnindis 17361
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 3836 . . . . . . 7  |-  ( x  e.  { (/) ,  A }  ->  ( x  =  (/)  \/  x  =  A ) )
2 topontop 16996 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
32ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  J  e.  Top )
4 0opn 16982 . . . . . . . . . 10  |-  ( J  e.  Top  ->  (/)  e.  J
)
53, 4syl 16 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  (/) 
e.  J )
6 imaeq2 5202 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( `' f " x )  =  ( `' f
" (/) ) )
7 ima0 5224 . . . . . . . . . . 11  |-  ( `' f " (/) )  =  (/)
86, 7syl6eq 2486 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( `' f " x )  =  (/) )
98eleq1d 2504 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ( `' f " x
)  e.  J  <->  (/)  e.  J
) )
105, 9syl5ibrcom 215 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  (/)  ->  ( `' f "
x )  e.  J
) )
11 fimacnv 5865 . . . . . . . . . . 11  |-  ( f : X --> A  -> 
( `' f " A )  =  X )
1211adantl 454 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  =  X )
13 toponmax 16998 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
1413ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  X  e.  J )
1512, 14eqeltrd 2512 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  e.  J
)
16 imaeq2 5202 . . . . . . . . . 10  |-  ( x  =  A  ->  ( `' f " x
)  =  ( `' f " A ) )
1716eleq1d 2504 . . . . . . . . 9  |-  ( x  =  A  ->  (
( `' f "
x )  e.  J  <->  ( `' f " A
)  e.  J ) )
1815, 17syl5ibrcom 215 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  A  ->  ( `' f
" x )  e.  J ) )
1910, 18jaod 371 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( ( x  =  (/)  \/  x  =  A )  ->  ( `' f " x )  e.  J ) )
201, 19syl5 31 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  e.  { (/)
,  A }  ->  ( `' f " x
)  e.  J ) )
2120ralrimiv 2790 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J )
2221ex 425 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J ) )
2322pm4.71d 617 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  <->  ( f : X --> A  /\  A. x  e.  { (/) ,  A }  ( `' f
" x )  e.  J ) ) )
24 id 21 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
25 elmapg 7034 . . . 4  |-  ( ( A  e.  V  /\  X  e.  J )  ->  ( f  e.  ( A  ^m  X )  <-> 
f : X --> A ) )
2624, 13, 25syl2anr 466 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( A  ^m  X )  <->  f : X
--> A ) )
27 indistopon 17070 . . . 4  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
28 iscn 17304 . . . 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 462 . . 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 279 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( J  Cn  { (/) ,  A } )  <->  f  e.  ( A  ^m  X ) ) )
3130eqrdv 2436 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  ( J  Cn  { (/) ,  A } )  =  ( A  ^m  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   (/)c0 3630   {cpr 3817   `'ccnv 4880   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   Topctop 16963  TopOnctopon 16964    Cn ccn 17293
This theorem is referenced by:  indishmph  17835  indistgp  18135  indispcon  24926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-top 16968  df-topon 16971  df-cn 17296
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