Theorem uhgrun 27425

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 27413	. . . . 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 2739	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		uhgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	uhgrf 27413	. . . . . 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 2745	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 4557	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	difeq1d 4060	. . . . . 6 ⊢ (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
15	14	feq3d 6583	. . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})))
16	10, 15	mpbird 256	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
17		uhgrun.i	. . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
18	5, 16, 17	fun2d 6634	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
19		uhgrun.un	. . . 4 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
20	19	dmeqd 5811	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
21		dmun 5816	. . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹)
22	20, 21	eqtrdi 2795	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23		uhgrun.v	. . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
24	23	pweqd 4557	. . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
25	24	difeq1d 4060	. . . 4 ⊢ (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
26	19, 22, 25	feq123d 6585	. . 3 ⊢ (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
27	18, 26	mpbird 256	. 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))
28		uhgrun.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
29		eqid 2739	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
30		eqid 2739	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
31	29, 30	isuhgr 27411	. . 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 256	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2109   ∖ cdif 3888   ∪ cun 3889   ∩ cin 3890  ∅c0 4261  𝒫 cpw 4538  {csn 4566  dom cdm 5588  ⟶wf 6426  ‘cfv 6430  Vtxcvtx 27347  iEdgciedg 27348  UHGraphcuhgr 27407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-id 5488  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-uhgr 27409

This theorem is referenced by:  uhgrunop  27426  ushgrun  27427