Theorem hmphtop 23786

Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
hmphtop	⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Proof of Theorem hmphtop

Step	Hyp	Ref	Expression
1		df-hmph 23764	. . 3 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6100	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23765	. . . . 5 ⊢ Homeo Fn (Top × Top)
4		fndm 6671	. . . . 5 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
5	3, 4	ax-mp 5	. . . 4 ⊢ dom Homeo = (Top × Top)
6	2, 5	sseqtri 4032	. . 3 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
7	1, 6	eqsstri 4030	. 2 ⊢ ≃ ⊆ (Top × Top)
8	7	brel 5750	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684  dom cdm 5685   “ cima 5688   Fn wfn 6556  1_oc1o 8499  Topctop 22899  Homeochmeo 23761   ≃ chmph 23762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-hmeo 23763  df-hmph 23764

This theorem is referenced by:  hmphtop1  23787  hmphtop2  23788