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Theorem cnmptid 20030
Description: The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
cnmptid.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
Assertion
Ref Expression
cnmptid  |-  ( ph  ->  ( x  e.  X  |->  x )  e.  ( J  Cn  J ) )
Distinct variable groups:    ph, x    x, J    x, X

Proof of Theorem cnmptid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 equcom 1743 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
21opabbii 4517 . . . . 5  |-  { <. x ,  y >.  |  x  =  y }  =  { <. x ,  y
>.  |  y  =  x }
3 dfid3 4802 . . . . 5  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
4 mptv 4545 . . . . 5  |-  ( x  e.  _V  |->  x )  =  { <. x ,  y >.  |  y  =  x }
52, 3, 43eqtr4i 2506 . . . 4  |-  _I  =  ( x  e.  _V  |->  x )
65reseq1i 5275 . . 3  |-  (  _I  |`  X )  =  ( ( x  e.  _V  |->  x )  |`  X )
7 ssv 3529 . . . 4  |-  X  C_  _V
8 resmpt 5329 . . . 4  |-  ( X 
C_  _V  ->  ( ( x  e.  _V  |->  x )  |`  X )  =  ( x  e.  X  |->  x ) )
97, 8ax-mp 5 . . 3  |-  ( ( x  e.  _V  |->  x )  |`  X )  =  ( x  e.  X  |->  x )
106, 9eqtri 2496 . 2  |-  (  _I  |`  X )  =  ( x  e.  X  |->  x )
11 cnmptid.j . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
12 idcn 19626 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X )  e.  ( J  Cn  J ) )
1311, 12syl 16 . 2  |-  ( ph  ->  (  _I  |`  X )  e.  ( J  Cn  J ) )
1410, 13syl5eqelr 2560 1  |-  ( ph  ->  ( x  e.  X  |->  x )  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   {copab 4510    |-> cmpt 4511    _I cid 4796    |` cres 5007   ` 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-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-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-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-top 19268  df-topon 19271  df-cn 19596
This theorem is referenced by:  xkoinjcn  20056  txcon  20058  imasnopn  20059  imasncld  20060  imasncls  20061  pt1hmeo  20175  istgp2  20458  tmdmulg  20459  tmdlactcn  20469  clsnsg  20476  tgpt0  20485  tlmtgp  20566  nmcn  21217  expcn  21244  divccn  21245  cncfmptid  21284  cdivcncf  21289  iirevcn  21298  iihalf1cn  21300  iihalf2cn  21302  icchmeo  21309  evth2  21328  pcocn  21385  pcopt  21390  pcopt2  21391  pcoass  21392  csscld  21557  clsocv  21558  dvcnvlem  22245  resqrtcn  22989  sqrtcn  22990  efrlim  23165  ipasslem7  25574  occllem  26044  hmopidmchi  26893  rmulccn  27735  cvxpcon  28512  cvmlift2lem2  28574  cvmlift2lem3  28575  cvmliftphtlem  28587
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