MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgreq12g Structured version   Visualization version   GIF version

Theorem uhgreq12g 28906
Description: If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
Hypotheses
Ref Expression
uhgrf.v 𝑉 = (Vtx‘𝐺)
uhgrf.e 𝐸 = (iEdg‘𝐺)
uhgreq12g.w 𝑊 = (Vtx‘𝐻)
uhgreq12g.f 𝐹 = (iEdg‘𝐻)
Assertion
Ref Expression
uhgreq12g (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))

Proof of Theorem uhgreq12g
StepHypRef Expression
1 uhgrf.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 uhgrf.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2isuhgr 28901 . . . 4 (𝐺𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
43adantr 479 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
54adantr 479 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
6 simpr 483 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → 𝐸 = 𝐹)
76dmeqd 5912 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8 pweq 4620 . . . . . 6 (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
98difeq1d 4121 . . . . 5 (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
109adantr 479 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
116, 7, 10feq123d 6716 . . 3 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
12 uhgreq12g.w . . . . . 6 𝑊 = (Vtx‘𝐻)
13 uhgreq12g.f . . . . . 6 𝐹 = (iEdg‘𝐻)
1412, 13isuhgr 28901 . . . . 5 (𝐻𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1514adantl 480 . . . 4 ((𝐺𝑋𝐻𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1615bicomd 222 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
1711, 16sylan9bbr 509 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
185, 17bitrd 278 1 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cdif 3946  c0 4326  𝒫 cpw 4606  {csn 4632  dom cdm 5682  wf 6549  cfv 6553  Vtxcvtx 28837  iEdgciedg 28838  UHGraphcuhgr 28897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-uhgr 28899
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator