Theorem hmpher 23288

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 23260	. . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6081	. . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23261	. . . . . 6 ⊢ Homeo Fn (Top × Top)
4	3	fndmi 6654	. . . . 5 ⊢ dom Homeo = (Top × Top)
5	2, 4	sseqtri 4019	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
6	1, 5	eqsstri 4017	. . 3 ⊢ ≃ ⊆ (Top × Top)
7		relxp 5695	. . 3 ⊢ Rel (Top × Top)
8		relss 5782	. . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃
10		hmphsym 23286	. 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
11		hmphtr 23287	. 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
12		hmphref 23285	. . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
13		hmphtop1 23283	. . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
14	12, 13	impbii 208	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
15	9, 10, 11, 14	iseri 8730	1 ⊢ ≃ Er Top

Syntax hints:   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3946   ⊆ wss 3949   class class class wbr 5149   × cxp 5675  ^◡ccnv 5676  dom cdm 5677   “ cima 5680  Rel wrel 5682  1_oc1o 8459   Er wer 8700  Topctop 22395  Homeochmeo 23257   ≃ chmph 23258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-hmeo 23259  df-hmph 23260

This theorem is referenced by:  ismntop  33006