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Theorem idhmeo 17766
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 17283 . 2  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X )  e.  ( J  Cn  J ) )
2 cnvresid 5490 . . 3  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
32, 1syl5eqel 2496 . 2  |-  ( J  e.  (TopOn `  X
)  ->  `' (  _I  |`  X )  e.  ( J  Cn  J
) )
4 ishmeo 17752 . 2  |-  ( (  _I  |`  X )  e.  ( J  Homeo  J )  <-> 
( (  _I  |`  X )  e.  ( J  Cn  J )  /\  `' (  _I  |`  X )  e.  ( J  Cn  J ) ) )
51, 3, 4sylanbrc 646 1  |-  ( J  e.  (TopOn `  X
)  ->  (  _I  |`  X )  e.  ( J  Homeo  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    _I cid 4461   `'ccnv 4844    |` cres 4847   ` cfv 5421  (class class class)co 6048  TopOnctopon 16922    Cn ccn 17250    Homeo chmeo 17746
This theorem is referenced by:  hmphref  17774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-top 16926  df-topon 16929  df-cn 17253  df-hmeo 17748
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