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Theorem hmpher 21965
 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 21937 . . . . . 6 ≃ = (Homeo “ (V ∖ 1o))
2 cnvimass 5730 . . . . . . 7 (Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3 hmeofn 21938 . . . . . . . 8 Homeo Fn (Top × Top)
4 fndm 6227 . . . . . . . 8 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . . . . 7 dom Homeo = (Top × Top)
62, 5sseqtri 3862 . . . . . 6 (Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
71, 6eqsstri 3860 . . . . 5 ≃ ⊆ (Top × Top)
8 relxp 5364 . . . . 5 Rel (Top × Top)
9 relss 5445 . . . . 5 ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
107, 8, 9mp2 9 . . . 4 Rel ≃
1110a1i 11 . . 3 (⊤ → Rel ≃ )
12 hmphsym 21963 . . . 4 (𝑥𝑦𝑦𝑥)
1312adantl 475 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
14 hmphtr 21964 . . . 4 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1514adantl 475 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
16 hmphref 21962 . . . . 5 (𝑥 ∈ Top → 𝑥𝑥)
17 hmphtop1 21960 . . . . 5 (𝑥𝑥𝑥 ∈ Top)
1816, 17impbii 201 . . . 4 (𝑥 ∈ Top ↔ 𝑥𝑥)
1918a1i 11 . . 3 (⊤ → (𝑥 ∈ Top ↔ 𝑥𝑥))
2011, 13, 15, 19iserd 8040 . 2 (⊤ → ≃ Er Top)
2120mptru 1664 1 ≃ Er Top
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1656  ⊤wtru 1657   ∈ wcel 2164  Vcvv 3414   ∖ cdif 3795   ⊆ wss 3798   class class class wbr 4875   × cxp 5344  ◡ccnv 5345  dom cdm 5346   “ cima 5349  Rel wrel 5351   Fn wfn 6122  1oc1o 7824   Er wer 8011  Topctop 21075  Homeochmeo 21934   ≃ chmph 21935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-1o 7831  df-er 8014  df-map 8129  df-top 21076  df-topon 21093  df-cn 21409  df-hmeo 21936  df-hmph 21937 This theorem is referenced by:  ismntop  30611
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