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Theorem cnconst 17012
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 5728 . . . 4  |-  ( B  e.  Y  ->  ( F : X --> { B } 
<->  F  =  ( X  X.  { B }
) ) )
21adantl 452 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F : X --> { B } 
<->  F  =  ( X  X.  { B }
) ) )
3 cnconst2 17011 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
433expa 1151 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( X  X.  { B }
)  e.  ( J  Cn  K ) )
5 eleq1 2343 . . . 4  |-  ( F  =  ( X  X.  { B } )  -> 
( F  e.  ( J  Cn  K )  <-> 
( X  X.  { B } )  e.  ( J  Cn  K ) ) )
64, 5syl5ibrcom 213 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F  =  ( X  X.  { B } )  ->  F  e.  ( J  Cn  K ) ) )
72, 6sylbid 206 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F : X --> { B }  ->  F  e.  ( J  Cn  K ) ) )
87impr 602 1  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  ( B  e.  Y  /\  F : X
--> { B } ) )  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  xrge0mulc1cn  23323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-topon 16639  df-cn 16957  df-cnp 16958
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