Theorem hmpher 23719

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 23691	. . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6038	. . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23692	. . . . . 6 ⊢ Homeo Fn (Top × Top)
4	3	fndmi 6593	. . . . 5 ⊢ dom Homeo = (Top × Top)
5	2, 4	sseqtri 3979	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
6	1, 5	eqsstri 3977	. . 3 ⊢ ≃ ⊆ (Top × Top)
7		relxp 5639	. . 3 ⊢ Rel (Top × Top)
8		relss 5728	. . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
9	6, 7, 8	mp2 9	. 2 ⊢ Rel ≃
10		hmphsym 23717	. 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
11		hmphtr 23718	. 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
12		hmphref 23716	. . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
13		hmphtop1 23714	. . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
14	12, 13	impbii 209	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
15	9, 10, 11, 14	iseri 8658	1 ⊢ ≃ Er Top

Syntax hints:   ∈ wcel 2113  Vcvv 3437   ∖ cdif 3895   ⊆ wss 3898   class class class wbr 5095   × cxp 5619  ^◡ccnv 5620  dom cdm 5621   “ cima 5624  Rel wrel 5626  1_oc1o 8387   Er wer 8628  Topctop 22828  Homeochmeo 23688   ≃ chmph 23689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-1o 8394  df-er 8631  df-map 8761  df-top 22829  df-topon 22846  df-cn 23162  df-hmeo 23690  df-hmph 23691

This theorem is referenced by:  ismntop  34111