Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmpher Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator