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Theorem cnconst 17272
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 5887 . . . 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 17271 . . . . 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 2449 . . . 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 1649    e. wcel 1717   {csn 3759    X. cxp 4818   -->wf 5392   ` cfv 5396  (class class class)co 6022  TopOnctopon 16884    Cn ccn 17212
This theorem is referenced by:  xrge0mulc1cn  24133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-map 6958  df-topgen 13596  df-top 16888  df-topon 16891  df-cn 17215  df-cnp 17216
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