Theorem hmphtop 23802

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 23780	. . 3 ⊢ ≃ = (◡Homeo “ (V ∖ 1o))
2		cnvimass 6102	. . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3		hmeofn 23781	. . . . 5 ⊢ Homeo Fn (Top × Top)
4		fndm 6672	. . . . 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 5754	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   class class class wbr 5148   × cxp 5687  ^◡ccnv 5688  dom cdm 5689   “ cima 5692   Fn wfn 6558  1_oc1o 8498  Topctop 22915  Homeochmeo 23777   ≃ chmph 23778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-hmeo 23779  df-hmph 23780

This theorem is referenced by:  hmphtop1  23803  hmphtop2  23804