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Theorem idhmeo 17564
Description: The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
idhmeo  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X )  e.  ( J  Homeo  J )
)

Proof of Theorem idhmeo
StepHypRef Expression
1 idcn 17087 . 2  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X )  e.  ( J  Cn  J ) )
2 cnvresid 5401 . . 3  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
32, 1syl5eqel 2442 . 2  |-  ( J  e.  (TopOn `  X
)  ->  `' (  _I  |`  X )  e.  ( J  Cn  J
) )
4 ishmeo 17550 . 2  |-  ( (  _I  |`  X )  e.  ( J  Homeo  J )  <-> 
( (  _I  |`  X )  e.  ( J  Cn  J )  /\  `' (  _I  |`  X )  e.  ( J  Cn  J ) ) )
51, 3, 4sylanbrc 645 1  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X )  e.  ( J  Homeo  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710    _I cid 4383   `'ccnv 4767    |` cres 4770   ` cfv 5334  (class class class)co 5942  TopOnctopon 16732    Cn ccn 17054    Homeo chmeo 17544
This theorem is referenced by:  hmphref  17572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-map 6859  df-top 16736  df-topon 16739  df-cn 17057  df-hmeo 17546
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