Theorem hmpher 23815

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 23787	. . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6113	. . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23788	. . . . . 6 ⊢ Homeo Fn (Top × Top)
4	3	fndmi 6685	. . . . 5 ⊢ dom Homeo = (Top × Top)
5	2, 4	sseqtri 4045	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
6	1, 5	eqsstri 4043	. . 3 ⊢ ≃ ⊆ (Top × Top)
7		relxp 5718	. . 3 ⊢ Rel (Top × Top)
8		relss 5805	. . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃
10		hmphsym 23813	. 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
11		hmphtr 23814	. 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
12		hmphref 23812	. . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
13		hmphtop1 23810	. . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
14	12, 13	impbii 209	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
15	9, 10, 11, 14	iseri 8792	1 ⊢ ≃ Er Top

Syntax hints:   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699  dom cdm 5700   “ cima 5703  Rel wrel 5705  1_oc1o 8517   Er wer 8762  Topctop 22922  Homeochmeo 23784   ≃ chmph 23785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-1o 8524  df-er 8765  df-map 8888  df-top 22923  df-topon 22940  df-cn 23258  df-hmeo 23786  df-hmph 23787

This theorem is referenced by:  ismntop  33974