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Theorem cnindis 19960
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 4036 . . . . . . 7  |-  ( x  e.  { (/) ,  A }  ->  ( x  =  (/)  \/  x  =  A ) )
2 topontop 19594 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
32ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  J  e.  Top )
4 0opn 19580 . . . . . . . . . 10  |-  ( J  e.  Top  ->  (/)  e.  J
)
53, 4syl 16 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  (/) 
e.  J )
6 imaeq2 5321 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( `' f " x )  =  ( `' f
" (/) ) )
7 ima0 5340 . . . . . . . . . . 11  |-  ( `' f " (/) )  =  (/)
86, 7syl6eq 2511 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( `' f " x )  =  (/) )
98eleq1d 2523 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ( `' f " x
)  e.  J  <->  (/)  e.  J
) )
105, 9syl5ibrcom 222 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  (/)  ->  ( `' f "
x )  e.  J
) )
11 fimacnv 5995 . . . . . . . . . . 11  |-  ( f : X --> A  -> 
( `' f " A )  =  X )
1211adantl 464 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  =  X )
13 toponmax 19596 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
1413ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  X  e.  J )
1512, 14eqeltrd 2542 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( `' f " A )  e.  J
)
16 imaeq2 5321 . . . . . . . . . 10  |-  ( x  =  A  ->  ( `' f " x
)  =  ( `' f " A ) )
1716eleq1d 2523 . . . . . . . . 9  |-  ( x  =  A  ->  (
( `' f "
x )  e.  J  <->  ( `' f " A
)  e.  J ) )
1815, 17syl5ibrcom 222 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  =  A  ->  ( `' f
" x )  e.  J ) )
1910, 18jaod 378 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( ( x  =  (/)  \/  x  =  A )  ->  ( `' f " x )  e.  J ) )
201, 19syl5 32 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  -> 
( x  e.  { (/)
,  A }  ->  ( `' f " x
)  e.  J ) )
2120ralrimiv 2866 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  /\  f : X --> A )  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J )
2221ex 432 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f : X --> A  ->  A. x  e.  { (/) ,  A }  ( `' f " x )  e.  J ) )
2322pm4.71d 632 . . 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 7425 . . . 4  |-  ( ( A  e.  V  /\  X  e.  J )  ->  ( f  e.  ( A  ^m  X )  <-> 
f : X --> A ) )
2624, 13, 25syl2anr 476 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( A  ^m  X )  <->  f : X
--> A ) )
27 indistopon 19669 . . . 4  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
28 iscn 19903 . . . 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 472 . . 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 286 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  (
f  e.  ( J  Cn  { (/) ,  A } )  <->  f  e.  ( A  ^m  X ) ) )
3130eqrdv 2451 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 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   (/)c0 3783   {cpr 4018   `'ccnv 4987   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Topctop 19561  TopOnctopon 19562    Cn ccn 19892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19566  df-topon 19569  df-cn 19895
This theorem is referenced by:  indishmph  20465  indistgp  20765  indispcon  28943
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