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.

Step	Hyp	Ref	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))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ dom Homeo = (Top × Top)
6	2, 5	sseqtri 3862	. . . . . 6 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
7	1, 6	eqsstri 3860	. . . . 5 ⊢ ≃ ⊆ (Top × Top)
8		relxp 5364	. . . . 5 ⊢ Rel (Top × Top)
9		relss 5445	. . . . 5 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
10	7, 8, 9	mp2 9	. . . 4 ⊢ Rel ≃
11	10	a1i 11	. . 3 ⊢ (⊤ → Rel ≃ )
12		hmphsym 21963	. . . 4 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
13	12	adantl 475	. . 3 ⊢ ((⊤ ∧ 𝑥 ≃ 𝑦) → 𝑦 ≃ 𝑥)
14		hmphtr 21964	. . . 4 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
15	14	adantl 475	. . 3 ⊢ ((⊤ ∧ (𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧)) → 𝑥 ≃ 𝑧)
16		hmphref 21962	. . . . 5 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
17		hmphtop1 21960	. . . . 5 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
18	16, 17	impbii 201	. . . 4 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
19	18	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥))
20	11, 13, 15, 19	iserd 8040	. 2 ⊢ (⊤ → ≃ Er Top)
21	20	mptru 1664	1 ⊢ ≃ Er Top

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  1_oc1o 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