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Theorem hmphref 22392
Description: "Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphref (𝐽 ∈ Top → 𝐽𝐽)

Proof of Theorem hmphref
StepHypRef Expression
1 toptopon2 21529 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2 idhmeo 22384 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → ( I ↾ 𝐽) ∈ (𝐽Homeo𝐽))
31, 2sylbi 220 . 2 (𝐽 ∈ Top → ( I ↾ 𝐽) ∈ (𝐽Homeo𝐽))
4 hmphi 22388 . 2 (( I ↾ 𝐽) ∈ (𝐽Homeo𝐽) → 𝐽𝐽)
53, 4syl 17 1 (𝐽 ∈ Top → 𝐽𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115   cuni 4824   class class class wbr 5052   I cid 5446  cres 5544  cfv 6343  (class class class)co 7149  Topctop 21504  TopOnctopon 21521  Homeochmeo 22364  chmph 22365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-1o 8098  df-map 8404  df-top 21505  df-topon 21522  df-cn 21838  df-hmeo 22366  df-hmph 22367
This theorem is referenced by:  hmpher  22395  hmph0  22406
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