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Theorem hmphtop 21802
Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmphtop (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Proof of Theorem hmphtop
StepHypRef Expression
1 df-hmph 21780 . . 3 ≃ = (Homeo “ (V ∖ 1𝑜))
2 cnvimass 5625 . . . 4 (Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo
3 hmeofn 21781 . . . . 5 Homeo Fn (Top × Top)
4 fndm 6129 . . . . 5 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . 4 dom Homeo = (Top × Top)
62, 5sseqtri 3786 . . 3 (Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top)
71, 6eqsstri 3784 . 2 ≃ ⊆ (Top × Top)
87brel 5307 1 (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720   class class class wbr 4787   × cxp 5248  ccnv 5249  dom cdm 5250  cima 5253   Fn wfn 6025  1𝑜c1o 7710  Topctop 20918  Homeochmeo 21777  chmph 21778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-hmeo 21779  df-hmph 21780
This theorem is referenced by:  hmphtop1  21803  hmphtop2  21804
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