Theorem hmpher 21587

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 21559	. . . . . 6 ⊢ ≃ = (◡Homeo “ (V ∖ 1𝑜))
2		cnvimass 5485	. . . . . . 7 ⊢ (◡Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo
3		hmeofn 21560	. . . . . . . 8 ⊢ Homeo Fn (Top × Top)
4		fndm 5990	. . . . . . . 8 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ dom Homeo = (Top × Top)
6	2, 5	sseqtri 3637	. . . . . 6 ⊢ (◡Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top)
7	1, 6	eqsstri 3635	. . . . 5 ⊢ ≃ ⊆ (Top × Top)
8		relxp 5227	. . . . 5 ⊢ Rel (Top × Top)
9		relss 5206	. . . . 5 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
10	7, 8, 9	mp2 9	. . . 4 ⊢ Rel ≃
11	10	a1i 11	. . 3 ⊢ (⊤ → Rel ≃ )
12		hmphsym 21585	. . . 4 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
13	12	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ≃ 𝑦) → 𝑦 ≃ 𝑥)
14		hmphtr 21586	. . . 4 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
15	14	adantl 482	. . 3 ⊢ ((⊤ ∧ (𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧)) → 𝑥 ≃ 𝑧)
16		hmphref 21584	. . . . 5 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
17		hmphtop1 21582	. . . . 5 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
18	16, 17	impbii 199	. . . 4 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
19	18	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥))
20	11, 13, 15, 19	iserd 7768	. 2 ⊢ (⊤ → ≃ Er Top)
21	20	trud 1493	1 ⊢ ≃ Er Top

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Rel wrel 5119   Fn wfn 5883  1_𝑜c1o 7553   Er wer 7739  Topctop 20698  Homeochmeo 21556   ≃ chmph 21557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558  df-hmph 21559

This theorem is referenced by:  ismntop  30070