Theorem hmphtop 22929

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 22907	. . 3 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 5989	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 22908	. . . . 5 ⊢ Homeo Fn (Top × Top)
4		fndm 6536	. . . . 5 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
5	3, 4	ax-mp 5	. . . 4 ⊢ dom Homeo = (Top × Top)
6	2, 5	sseqtri 3957	. . 3 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
7	1, 6	eqsstri 3955	. 2 ⊢ ≃ ⊆ (Top × Top)
8	7	brel 5652	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   class class class wbr 5074   × cxp 5587  ^◡ccnv 5588  dom cdm 5589   “ cima 5592   Fn wfn 6428  1_oc1o 8290  Topctop 22042  Homeochmeo 22904   ≃ chmph 22905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-hmeo 22906  df-hmph 22907

This theorem is referenced by:  hmphtop1  22930  hmphtop2  22931