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Theorem hmpher 22392
 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 22364 . . . 4 ≃ = (Homeo “ (V ∖ 1o))
2 cnvimass 5936 . . . . 5 (Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3 hmeofn 22365 . . . . . 6 Homeo Fn (Top × Top)
43fndmi 6444 . . . . 5 dom Homeo = (Top × Top)
52, 4sseqtri 3989 . . . 4 (Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
61, 5eqsstri 3987 . . 3 ≃ ⊆ (Top × Top)
7 relxp 5560 . . 3 Rel (Top × Top)
8 relss 5643 . . 3 ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
96, 7, 8mp2 9 . 2 Rel ≃
10 hmphsym 22390 . 2 (𝑥𝑦𝑦𝑥)
11 hmphtr 22391 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
12 hmphref 22389 . . 3 (𝑥 ∈ Top → 𝑥𝑥)
13 hmphtop1 22387 . . 3 (𝑥𝑥𝑥 ∈ Top)
1412, 13impbii 212 . 2 (𝑥 ∈ Top ↔ 𝑥𝑥)
159, 10, 11, 14iseri 8312 1 ≃ Er Top
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  Vcvv 3480   ∖ cdif 3916   ⊆ wss 3919   class class class wbr 5052   × cxp 5540  ◡ccnv 5541  dom cdm 5542   “ cima 5545  Rel wrel 5547  1oc1o 8091   Er wer 8282  Topctop 21501  Homeochmeo 22361   ≃ chmph 22362 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-er 8285  df-map 8404  df-top 21502  df-topon 21519  df-cn 21835  df-hmeo 22363  df-hmph 22364 This theorem is referenced by:  ismntop  31324
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