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Theorem cnconst 19544
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 6106 . . . 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 19543 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y
)  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
433expa 1191 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( X  X.  { B }
)  e.  ( J  Cn  K ) )
5 eleq1 2532 . . . 4  |-  ( F  =  ( X  X.  { B } )  -> 
( F  e.  ( J  Cn  K )  <-> 
( X  X.  { B } )  e.  ( J  Cn  K ) ) )
64, 5syl5ibrcom 222 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  B  e.  Y )  ->  ( F  =  ( X  X.  { B } )  ->  F  e.  ( J  Cn  K ) ) )
72, 6sylbid 215 . 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 184    /\ wa 369    = wceq 1374    e. wcel 1762   {csn 4020    X. cxp 4990   -->wf 5575   ` cfv 5579  (class class class)co 6275  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-topgen 14688  df-top 19159  df-topon 19162  df-cn 19487  df-cnp 19488
This theorem is referenced by:  xrge0mulc1cn  27545
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