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Theorem hmpher 21497
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 21469 . . . . . 6 ≃ = (Homeo “ (V ∖ 1𝑜))
2 cnvimass 5444 . . . . . . 7 (Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo
3 hmeofn 21470 . . . . . . . 8 Homeo Fn (Top × Top)
4 fndm 5948 . . . . . . . 8 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . . . . 7 dom Homeo = (Top × Top)
62, 5sseqtri 3616 . . . . . 6 (Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top)
71, 6eqsstri 3614 . . . . 5 ≃ ⊆ (Top × Top)
8 relxp 5188 . . . . 5 Rel (Top × Top)
9 relss 5167 . . . . 5 ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
107, 8, 9mp2 9 . . . 4 Rel ≃
1110a1i 11 . . 3 (⊤ → Rel ≃ )
12 hmphsym 21495 . . . 4 (𝑥𝑦𝑦𝑥)
1312adantl 482 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
14 hmphtr 21496 . . . 4 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1514adantl 482 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
16 hmphref 21494 . . . . 5 (𝑥 ∈ Top → 𝑥𝑥)
17 hmphtop1 21492 . . . . 5 (𝑥𝑥𝑥 ∈ Top)
1816, 17impbii 199 . . . 4 (𝑥 ∈ Top ↔ 𝑥𝑥)
1918a1i 11 . . 3 (⊤ → (𝑥 ∈ Top ↔ 𝑥𝑥))
2011, 13, 15, 19iserd 7713 . 2 (⊤ → ≃ Er Top)
2120trud 1490 1 ≃ Er Top
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  Vcvv 3186  cdif 3552  wss 3555   class class class wbr 4613   × cxp 5072  ccnv 5073  dom cdm 5074  cima 5077  Rel wrel 5079   Fn wfn 5842  1𝑜c1o 7498   Er wer 7684  Topctop 20617  Homeochmeo 21466  chmph 21467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-1o 7505  df-er 7687  df-map 7804  df-top 20621  df-topon 20623  df-cn 20941  df-hmeo 21468  df-hmph 21469
This theorem is referenced by:  ismntop  29852
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