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Theorem cnindis 17348
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 3826 . . . . . . 7  |-  ( x  e.  { (/) ,  A }  ->  ( x  =  (/)  \/  x  =  A ) )
2 topontop 16983 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
32ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  J  e.  Top )
4 0opn 16969 . . . . . . . . . 10  |-  ( J  e.  Top  ->  (/)  e.  J
)
53, 4syl 16 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  (/) 
e.  J )
6 imaeq2 5191 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( `' f " x )  =  ( `' f
" (/) ) )
7 ima0 5213 . . . . . . . . . . 11  |-  ( `' f " (/) )  =  (/)
86, 7syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( `' f " x )  =  (/) )
98eleq1d 2501 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ( `' f " x
)  e.  J  <->  (/)  e.  J
) )
105, 9syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  (/)  ->  ( `' f "
x )  e.  J
) )
11 fimacnv 5854 . . . . . . . . . . 11  |-  ( f : X --> A  -> 
( `' f " A )  =  X )
1211adantl 453 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  =  X )
13 toponmax 16985 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
1413ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  X  e.  J )
1512, 14eqeltrd 2509 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  e.  J
)
16 imaeq2 5191 . . . . . . . . . 10  |-  ( x  =  A  ->  ( `' f " x
)  =  ( `' f " A ) )
1716eleq1d 2501 . . . . . . . . 9  |-  ( x  =  A  ->  (
( `' f "
x )  e.  J  <->  ( `' f " A
)  e.  J ) )
1815, 17syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  A  ->  ( `' f
" x )  e.  J ) )
1910, 18jaod 370 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( ( x  =  (/)  \/  x  =  A )  ->  ( `' f " x )  e.  J ) )
201, 19syl5 30 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  e.  { (/)
,  A }  ->  ( `' f " x
)  e.  J ) )
2120ralrimiv 2780 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J )
2221ex 424 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J ) )
2322pm4.71d 616 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  <->  ( f : X --> A  /\  A. x  e.  { (/) ,  A }  ( `' f
" x )  e.  J ) ) )
24 id 20 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
25 elmapg 7023 . . . 4  |-  ( ( A  e.  V  /\  X  e.  J )  ->  ( f  e.  ( A  ^m  X )  <-> 
f : X --> A ) )
2624, 13, 25syl2anr 465 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( A  ^m  X )  <->  f : X
--> A ) )
27 indistopon 17057 . . . 4  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
28 iscn 17291 . . . 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 461 . . 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 278 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( J  Cn  { (/) ,  A } )  <->  f  e.  ( A  ^m  X ) ) )
3130eqrdv 2433 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 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   (/)c0 3620   {cpr 3807   `'ccnv 4869   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   Topctop 16950  TopOnctopon 16951    Cn ccn 17280
This theorem is referenced by:  indishmph  17822  indistgp  18122  indispcon  24913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-top 16955  df-topon 16958  df-cn 17283
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