Theorem hmpher 23508

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.

Step	Hyp	Ref	Expression
1		df-hmph 23480	. . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6080	. . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23481	. . . . . 6 ⊢ Homeo Fn (Top × Top)
4	3	fndmi 6653	. . . . 5 ⊢ dom Homeo = (Top × Top)
5	2, 4	sseqtri 4018	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
6	1, 5	eqsstri 4016	. . 3 ⊢ ≃ ⊆ (Top × Top)
7		relxp 5694	. . 3 ⊢ Rel (Top × Top)
8		relss 5781	. . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃
10		hmphsym 23506	. 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
11		hmphtr 23507	. 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
12		hmphref 23505	. . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
13		hmphtop1 23503	. . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
14	12, 13	impbii 208	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
15	9, 10, 11, 14	iseri 8732	1 ⊢ ≃ Er Top

Syntax hints:   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948   class class class wbr 5148   × cxp 5674  ^◡ccnv 5675  dom cdm 5676   “ cima 5679  Rel wrel 5681  1_oc1o 8461   Er wer 8702  Topctop 22615  Homeochmeo 23477   ≃ chmph 23478

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-map 8824  df-top 22616  df-topon 22633  df-cn 22951  df-hmeo 23479  df-hmph 23480

This theorem is referenced by:  ismntop  33292