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Theorem uhgreq12g 26533
 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 26528 . . . 4 (𝐺𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
43adantr 481 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
54adantr 481 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
6 simpr 485 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → 𝐸 = 𝐹)
76dmeqd 5660 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8 pweq 4456 . . . . . 6 (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
98difeq1d 4019 . . . . 5 (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
109adantr 481 . . . 4 ((𝑉 = 𝑊𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
116, 7, 10feq123d 6371 . . 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 26528 . . . . 5 (𝐻𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1514adantl 482 . . . 4 ((𝐺𝑋𝐻𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
1615bicomd 224 . . 3 ((𝐺𝑋𝐻𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
1711, 16sylan9bbr 511 . 2 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
185, 17bitrd 280 1 (((𝐺𝑋𝐻𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ∖ cdif 3856  ∅c0 4211  𝒫 cpw 4453  {csn 4472  dom cdm 5443  ⟶wf 6221  ‘cfv 6225  Vtxcvtx 26464  iEdgciedg 26465  UHGraphcuhgr 26524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5101 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-uhgr 26526 This theorem is referenced by: (None)
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