Theorem hmpher 23906

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 23878	. . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6082	. . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23879	. . . . . 6 ⊢ Homeo Fn (Top × Top)
4	3	fndmi 6637	. . . . 5 ⊢ dom Homeo = (Top × Top)
5	2, 4	sseqtri 3993	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
6	1, 5	eqsstri 3991	. . 3 ⊢ ≃ ⊆ (Top × Top)
7		relxp 5677	. . 3 ⊢ Rel (Top × Top)
8		relss 5766	. . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃
10		hmphsym 23904	. 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
11		hmphtr 23905	. 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
12		hmphref 23903	. . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
13		hmphtop1 23901	. . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
14	12, 13	impbii 212	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
15	9, 10, 11, 14	iseri 8718	1 ⊢ ≃ Er Top

Syntax hints:   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913   class class class wbr 5110   × cxp 5657  ^◡ccnv 5658  dom cdm 5659   “ cima 5662  Rel wrel 5664  1_oc1o 8442   Er wer 8687  Topctop 23015  Homeochmeo 23875   ≃ chmph 23876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-1o 8449  df-er 8690  df-map 8822  df-top 23016  df-topon 23033  df-cn 23349  df-hmeo 23877  df-hmph 23878

This theorem is referenced by:  ismntop  34357