Theorem hmpher 21397

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 21369	. . . . . 6 ⊢ ≃ = (◡Homeo “ (V ∖ 1𝑜))
2		cnvimass 5404	. . . . . . 7 ⊢ (◡Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo
3		hmeofn 21370	. . . . . . . 8 ⊢ Homeo Fn (Top × Top)
4		fndm 5904	. . . . . . . 8 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ dom Homeo = (Top × Top)
6	2, 5	sseqtri 3600	. . . . . 6 ⊢ (◡Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top)
7	1, 6	eqsstri 3598	. . . . 5 ⊢ ≃ ⊆ (Top × Top)
8		relxp 5150	. . . . 5 ⊢ Rel (Top × Top)
9		relss 5129	. . . . 5 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
10	7, 8, 9	mp2 9	. . . 4 ⊢ Rel ≃
11	10	a1i 11	. . 3 ⊢ (⊤ → Rel ≃ )
12		hmphsym 21395	. . . 4 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
13	12	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ≃ 𝑦) → 𝑦 ≃ 𝑥)
14		hmphtr 21396	. . . 4 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
15	14	adantl 481	. . 3 ⊢ ((⊤ ∧ (𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧)) → 𝑥 ≃ 𝑧)
16		hmphref 21394	. . . . 5 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
17		hmphtop1 21392	. . . . 5 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
18	16, 17	impbii 198	. . . 4 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
19	18	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥))
20	11, 13, 15, 19	iserd 7655	. 2 ⊢ (⊤ → ≃ Er Top)
21	20	trud 1484	1 ⊢ ≃ Er Top

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Rel wrel 5043   Fn wfn 5799  1_𝑜c1o 7440   Er wer 7626  Topctop 20517  Homeochmeo 21366   ≃ chmph 21367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-hmeo 21368  df-hmph 21369

This theorem is referenced by:  ismntop  29398