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Theorem cnmptc 20675
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 4897 . 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 20297 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  Y
)  ->  ( X  X.  { P } )  e.  ( J  Cn  K ) )
62, 3, 4, 5syl3anc 1264 . 2  |-  ( ph  ->  ( X  X.  { P } )  e.  ( J  Cn  K ) )
71, 6syl5eqelr 2512 1  |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1872   {csn 3998    |-> cmpt 4482    X. cxp 4851   ` cfv 5601  (class class class)co 6305  TopOnctopon 19916    Cn ccn 20238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7485  df-topgen 15341  df-top 19919  df-topon 19921  df-cn 20241  df-cnp 20242
This theorem is referenced by:  cnmpt2c  20683  xkoinjcn  20700  txcon  20702  imasnopn  20703  imasncld  20704  imasncls  20705  istgp2  21104  tmdmulg  21105  tmdgsum  21108  tmdlactcn  21115  clsnsg  21122  tgpt0  21131  tlmtgp  21208  nmcn  21860  fsumcn  21900  expcn  21902  divccn  21903  cncfmptc  21941  cdivcncf  21947  iirevcn  21956  iihalf1cn  21958  iihalf2cn  21960  icchmeo  21967  evth  21985  evth2  21986  pcocn  22046  pcopt  22051  pcopt2  22052  pcoass  22053  csscld  22218  clsocv  22219  dvcnvlem  22926  plycn  23213  psercn2  23376  resqrtcn  23687  sqrtcn  23688  atansopn  23856  efrlim  23893  ipasslem7  26475  occllem  26954  rmulccn  28742  txsconlem  29971  cvxpcon  29973  cvmlift2lem2  30035  cvmlift2lem3  30036  cvmliftphtlem  30048  sinccvglem  30324  areacirclem2  31997
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