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Theorem cnconst 17338
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 5938 . . . 4  |-  ( B  e.  Y  ->  ( F : X --> { B } 
<->  F  =  ( X  X.  { B }
) ) )
21adantl 453 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F : X --> { B } 
<->  F  =  ( X  X.  { B }
) ) )
3 cnconst2 17337 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
433expa 1153 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( X  X.  { B }
)  e.  ( J  Cn  K ) )
5 eleq1 2495 . . . 4  |-  ( F  =  ( X  X.  { B } )  -> 
( F  e.  ( J  Cn  K )  <-> 
( X  X.  { B } )  e.  ( J  Cn  K ) ) )
64, 5syl5ibrcom 214 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F  =  ( X  X.  { B } )  ->  F  e.  ( J  Cn  K ) ) )
72, 6sylbid 207 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F : X --> { B }  ->  F  e.  ( J  Cn  K ) ) )
87impr 603 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073  TopOnctopon 16949    Cn ccn 17278
This theorem is referenced by:  xrge0mulc1cn  24317
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-csb 3244  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-iun 4087  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-1st 6341  df-2nd 6342  df-map 7012  df-topgen 13657  df-top 16953  df-topon 16956  df-cn 17281  df-cnp 17282
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