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Theorem cnmptc 20289
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 5052 . 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 19911 . . 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 2550 1  |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819   {csn 4032    |-> cmpt 4515    X. cxp 5006   ` cfv 5594  (class class class)co 6296  TopOnctopon 19522    Cn ccn 19852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-topgen 14861  df-top 19526  df-topon 19529  df-cn 19855  df-cnp 19856
This theorem is referenced by:  cnmpt2c  20297  xkoinjcn  20314  txcon  20316  imasnopn  20317  imasncld  20318  imasncls  20319  istgp2  20716  tmdmulg  20717  tmdgsum  20720  tmdlactcn  20727  clsnsg  20734  tgpt0  20743  tlmtgp  20824  nmcn  21475  fsumcn  21500  expcn  21502  divccn  21503  cncfmptc  21541  cdivcncf  21547  iirevcn  21556  iihalf1cn  21558  iihalf2cn  21560  icchmeo  21567  evth  21585  evth2  21586  pcocn  21643  pcopt  21648  pcopt2  21649  pcoass  21650  csscld  21815  clsocv  21816  dvcnvlem  22503  plycn  22784  psercn2  22944  resqrtcn  23249  sqrtcn  23250  atansopn  23389  efrlim  23425  ipasslem7  25878  occllem  26348  rmulccn  28071  txsconlem  28882  cvxpcon  28884  cvmlift2lem2  28946  cvmlift2lem3  28947  cvmliftphtlem  28959  sinccvglem  29235  areacirclem2  30313
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