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	⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))

Proof of Theorem uhgreq12g

Step	Hyp	Ref	Expression
1		uhgrf.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrf.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isuhgr 26528	. . . 4 ⊢ (𝐺 ∈ 𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4	3	adantr 481	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
5	4	adantr 481	. 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
6		simpr 485	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐸 = 𝐹)
7	6	dmeqd 5660	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → dom 𝐸 = dom 𝐹)
8		pweq 4456	. . . . . 6 ⊢ (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊)
9	8	difeq1d 4019	. . . . 5 ⊢ (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
10	9	adantr 481	. . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅}))
11	6, 7, 10	feq123d 6371	. . 3 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 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)
14	12, 13	isuhgr 26528	. . . . 5 ⊢ (𝐻 ∈ 𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
15	14	adantl 482	. . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅})))
16	15	bicomd 224	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
17	11, 16	sylan9bbr 511	. 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph))
18	5, 17	bitrd 280	1 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph))

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)