Theorem hmpher 22392

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 22364	. . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 5949	. . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 22365	. . . . . 6 ⊢ Homeo Fn (Top × Top)
4		fndm 6455	. . . . . 6 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
5	3, 4	ax-mp 5	. . . . 5 ⊢ dom Homeo = (Top × Top)
6	2, 5	sseqtri 4003	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
7	1, 6	eqsstri 4001	. . 3 ⊢ ≃ ⊆ (Top × Top)
8		relxp 5573	. . 3 ⊢ Rel (Top × Top)
9		relss 5656	. . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ ))
10	7, 8, 9	mp2 9	. 2 ⊢ Rel ≃
11		hmphsym 22390	. 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥)
12		hmphtr 22391	. 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧)
13		hmphref 22389	. . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥)
14		hmphtop1 22387	. . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top)
15	13, 14	impbii 211	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥)
16	10, 11, 12, 15	iseri 8316	1 ⊢ ≃ Er Top

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936   class class class wbr 5066   × cxp 5553  ^◡ccnv 5554  dom cdm 5555   “ cima 5558  Rel wrel 5560   Fn wfn 6350  1_oc1o 8095   Er wer 8286  Topctop 21501  Homeochmeo 22361   ≃ chmph 22362

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-1o 8102  df-er 8289  df-map 8408  df-top 21502  df-topon 21519  df-cn 21835  df-hmeo 22363  df-hmph 22364

This theorem is referenced by:  ismntop  31267