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Theorem cnconst 20080
Description: A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
cnconst  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( B  e.  Y  /\  F : X
--> { B } ) )  ->  F  e.  ( J  Cn  K
) )

Proof of Theorem cnconst
StepHypRef Expression
1 fconst2g 6108 . . . 4  |-  ( B  e.  Y  ->  ( F : X --> { B } 
<->  F  =  ( X  X.  { B }
) ) )
21adantl 466 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F : X --> { B } 
<->  F  =  ( X  X.  { B }
) ) )
3 cnconst2 20079 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
433expa 1199 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( X  X.  { B }
)  e.  ( J  Cn  K ) )
5 eleq1 2476 . . . 4  |-  ( F  =  ( X  X.  { B } )  -> 
( F  e.  ( J  Cn  K )  <-> 
( X  X.  { B } )  e.  ( J  Cn  K ) ) )
64, 5syl5ibrcom 224 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F  =  ( X  X.  { B } )  ->  F  e.  ( J  Cn  K ) ) )
72, 6sylbid 217 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F : X --> { B }  ->  F  e.  ( J  Cn  K ) ) )
87impr 619 1  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( B  e.  Y  /\  F : X
--> { B } ) )  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   {csn 3974    X. cxp 4823   -->wf 5567   ` cfv 5571  (class class class)co 6280  TopOnctopon 19689    Cn ccn 20020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-map 7461  df-topgen 15060  df-top 19693  df-topon 19696  df-cn 20023  df-cnp 20024
This theorem is referenced by:  xrge0mulc1cn  28389  cxpcncf2  37084
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