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Theorem hmpher 23815
Description: "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmpher ≃ Er Top

Proof of Theorem hmpher
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hmph 23787 . . . 4 ≃ = (Homeo “ (V ∖ 1o))
2 cnvimass 6113 . . . . 5 (Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3 hmeofn 23788 . . . . . 6 Homeo Fn (Top × Top)
43fndmi 6685 . . . . 5 dom Homeo = (Top × Top)
52, 4sseqtri 4045 . . . 4 (Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
61, 5eqsstri 4043 . . 3 ≃ ⊆ (Top × Top)
7 relxp 5718 . . 3 Rel (Top × Top)
8 relss 5805 . . 3 ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
96, 7, 8mp2 9 . 2 Rel ≃
10 hmphsym 23813 . 2 (𝑥𝑦𝑦𝑥)
11 hmphtr 23814 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
12 hmphref 23812 . . 3 (𝑥 ∈ Top → 𝑥𝑥)
13 hmphtop1 23810 . . 3 (𝑥𝑥𝑥 ∈ Top)
1412, 13impbii 209 . 2 (𝑥 ∈ Top ↔ 𝑥𝑥)
159, 10, 11, 14iseri 8792 1 ≃ Er Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3488  cdif 3973  wss 3976   class class class wbr 5166   × cxp 5698  ccnv 5699  dom cdm 5700  cima 5703  Rel wrel 5705  1oc1o 8517   Er wer 8762  Topctop 22922  Homeochmeo 23784  chmph 23785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-1o 8524  df-er 8765  df-map 8888  df-top 22923  df-topon 22940  df-cn 23258  df-hmeo 23786  df-hmph 23787
This theorem is referenced by:  ismntop  33974
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