Theorem uhgrun 25969

Description: The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
uhgrun.g	⊢ (𝜑 → 𝐺 ∈ UHGraph )
uhgrun.h	⊢ (𝜑 → 𝐻 ∈ UHGraph )
uhgrun.e	⊢ 𝐸 = (iEdg‘𝐺)
uhgrun.f	⊢ 𝐹 = (iEdg‘𝐻)
uhgrun.vg	⊢ 𝑉 = (Vtx‘𝐺)
uhgrun.vh	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
uhgrun.i	⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
uhgrun.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
uhgrun.v	⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
uhgrun.un	⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))

Assertion

Ref	Expression
uhgrun	⊢ (𝜑 → 𝑈 ∈ UHGraph )

Proof of Theorem uhgrun

Step	Hyp	Ref	Expression
1		uhgrun.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ UHGraph )
2		uhgrun.vg	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3		uhgrun.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	uhgrf 25957	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
6		uhgrun.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ UHGraph )
7		eqid 2622	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		uhgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	uhgrf 25957	. . . . . 6 ⊢ (𝐻 ∈ UHGraph → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
10	6, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
11		uhgrun.vh	. . . . . . . . 9 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
12	11	eqcomd 2628	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 4163	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	difeq1d 3727	. . . . . 6 ⊢ (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
15	14	feq3d 6032	. . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})))
16	10, 15	mpbird 247	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
17		uhgrun.i	. . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
18	5, 16, 17	fun2d 6068	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
19		uhgrun.un	. . . 4 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
20	19	dmeqd 5326	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
21		dmun 5331	. . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹)
22	20, 21	syl6eq 2672	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23		uhgrun.v	. . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
24	23	pweqd 4163	. . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
25	24	difeq1d 3727	. . . 4 ⊢ (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
26	19, 22, 25	feq123d 6034	. . 3 ⊢ (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
27	18, 26	mpbird 247	. 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))
28		uhgrun.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
29		eqid 2622	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
30		eqid 2622	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
31	29, 30	isuhgr 25955	. . 3 ⊢ (𝑈 ∈ 𝑊 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
32	28, 31	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
33	27, 32	mpbird 247	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph )

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  𝒫 cpw 4158  {csn 4177  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-uhgr 25953

This theorem is referenced by:  uhgrunop  25970  ushgrun  25971