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Theorem cnmptc 20031
Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptid.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
cnmptc.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
cnmptc.p  |-  ( ph  ->  P  e.  Y )
Assertion
Ref Expression
cnmptc  |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
Distinct variable groups:    ph, x    x, J    x, X    x, Y    x, K    x, P

Proof of Theorem cnmptc
StepHypRef Expression
1 fconstmpt 5049 . 2  |-  ( X  X.  { P }
)  =  ( x  e.  X  |->  P )
2 cnmptid.j . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3 cnmptc.k . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
4 cnmptc.p . . 3  |-  ( ph  ->  P  e.  Y )
5 cnconst2 19652 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  Y
)  ->  ( X  X.  { P } )  e.  ( J  Cn  K ) )
62, 3, 4, 5syl3anc 1228 . 2  |-  ( ph  ->  ( X  X.  { P } )  e.  ( J  Cn  K ) )
71, 6syl5eqelr 2560 1  |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   {csn 4033    |-> cmpt 4511    X. cxp 5003   ` cfv 5594  (class class class)co 6295  TopOnctopon 19264    Cn ccn 19593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-topgen 14716  df-top 19268  df-topon 19271  df-cn 19596  df-cnp 19597
This theorem is referenced by:  cnmpt2c  20039  xkoinjcn  20056  txcon  20058  imasnopn  20059  imasncld  20060  imasncls  20061  istgp2  20458  tmdmulg  20459  tmdgsum  20462  tmdlactcn  20469  clsnsg  20476  tgpt0  20485  tlmtgp  20566  nmcn  21217  fsumcn  21242  expcn  21244  divccn  21245  cncfmptc  21283  cdivcncf  21289  iirevcn  21298  iihalf1cn  21300  iihalf2cn  21302  icchmeo  21309  evth  21327  evth2  21328  pcocn  21385  pcopt  21390  pcopt2  21391  pcoass  21392  csscld  21557  clsocv  21558  dvcnvlem  22245  plycn  22525  psercn2  22685  resqrtcn  22989  sqrtcn  22990  atansopn  23129  efrlim  23165  ipasslem7  25565  occllem  26035  rmulccn  27726  txsconlem  28501  cvxpcon  28503  cvmlift2lem2  28565  cvmlift2lem3  28566  cvmliftphtlem  28578  sinccvglem  28854  areacirclem2  30026
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