Theorem hmphref 22383

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 → 𝐽 ≃ 𝐽)

Proof of Theorem hmphref

Step	Hyp	Ref	Expression
1		toptopon2 21520	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2		idhmeo 22375	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → ( I ↾ ∪ 𝐽) ∈ (𝐽Homeo𝐽))
3	1, 2	sylbi 219	. 2 ⊢ (𝐽 ∈ Top → ( I ↾ ∪ 𝐽) ∈ (𝐽Homeo𝐽))
4		hmphi 22379	. 2 ⊢ (( I ↾ ∪ 𝐽) ∈ (𝐽Homeo𝐽) → 𝐽 ≃ 𝐽)
5	3, 4	syl 17	1 ⊢ (𝐽 ∈ Top → 𝐽 ≃ 𝐽)

Syntax hints:   → wi 4   ∈ wcel 2110  ∪ cuni 4832   class class class wbr 5059   I cid 5454   ↾ cres 5552  ‘cfv 6350  (class class class)co 7150  Topctop 21495  TopOnctopon 21512  Homeochmeo 22355   ≃ chmph 22356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-1o 8096  df-map 8402  df-top 21496  df-topon 21513  df-cn 21829  df-hmeo 22357  df-hmph 22358

This theorem is referenced by:  hmpher  22386  hmph0  22397