Theorem uhgraun 25840

Description: The union of two (undirected) hypergraphs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are hypergraphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices), analogous to umgraun 25857. (Contributed by Alexander van der Vekens, 27-Dec-2017.)

Hypotheses

Ref	Expression
uhgraun.e	⊢ (𝜑 → 𝐸 Fn 𝐴)
uhgraun.f	⊢ (𝜑 → 𝐹 Fn 𝐵)
uhgraun.i	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
uhgraun.ge	⊢ (𝜑 → 𝑉 UHGrph 𝐸)
uhgraun.gf	⊢ (𝜑 → 𝑉 UHGrph 𝐹)

Assertion

Ref	Expression
uhgraun	⊢ (𝜑 → 𝑉 UHGrph (𝐸 ∪ 𝐹))

Proof of Theorem uhgraun

Step	Hyp	Ref	Expression
1		uhgraun.ge	. . . . . 6 ⊢ (𝜑 → 𝑉 UHGrph 𝐸)
2		uhgraf 25828	. . . . . 6 ⊢ (𝑉 UHGrph 𝐸 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
4		uhgraun.e	. . . . 5 ⊢ (𝜑 → 𝐸 Fn 𝐴)
5		fndm 5904	. . . . . . 7 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
6	5	feq2d 5944	. . . . . 6 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 𝑉 ∖ {∅})))
7	6	biimpac 502	. . . . 5 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 𝑉 ∖ {∅}))
8	3, 4, 7	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐸:𝐴⟶(𝒫 𝑉 ∖ {∅}))
9		uhgraun.gf	. . . . . 6 ⊢ (𝜑 → 𝑉 UHGrph 𝐹)
10		uhgraf 25828	. . . . . 6 ⊢ (𝑉 UHGrph 𝐹 → 𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
12		uhgraun.f	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐵)
13		fndm 5904	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
14	13	feq2d 5944	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:𝐵⟶(𝒫 𝑉 ∖ {∅})))
15	14	biimpac 502	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝐹 Fn 𝐵) → 𝐹:𝐵⟶(𝒫 𝑉 ∖ {∅}))
16	11, 12, 15	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(𝒫 𝑉 ∖ {∅}))
17		uhgraun.i	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
18		fun2 5980	. . . 4 ⊢ (((𝐸:𝐴⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝐹:𝐵⟶(𝒫 𝑉 ∖ {∅})) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶(𝒫 𝑉 ∖ {∅}))
19	8, 16, 17, 18	syl21anc 1317	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶(𝒫 𝑉 ∖ {∅}))
20		fdm 5964	. . . . 5 ⊢ ((𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶(𝒫 𝑉 ∖ {∅}) → dom (𝐸 ∪ 𝐹) = (𝐴 ∪ 𝐵))
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → dom (𝐸 ∪ 𝐹) = (𝐴 ∪ 𝐵))
22	21	feq2d 5944	. . 3 ⊢ (𝜑 → ((𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶(𝒫 𝑉 ∖ {∅}) ↔ (𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶(𝒫 𝑉 ∖ {∅})))
23	19, 22	mpbird 246	. 2 ⊢ (𝜑 → (𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
24		reluhgra 25823	. . . 4 ⊢ Rel UHGrph
25		releldm 5279	. . . 4 ⊢ ((Rel UHGrph ∧ 𝑉 UHGrph 𝐸) → 𝑉 ∈ dom UHGrph )
26	24, 1, 25	sylancr 694	. . 3 ⊢ (𝜑 → 𝑉 ∈ dom UHGrph )
27		uhgrav 25825	. . . . . 6 ⊢ (𝑉 UHGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
28	27	simprd 478	. . . . 5 ⊢ (𝑉 UHGrph 𝐸 → 𝐸 ∈ V)
29	1, 28	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
30		uhgrav 25825	. . . . . 6 ⊢ (𝑉 UHGrph 𝐹 → (𝑉 ∈ V ∧ 𝐹 ∈ V))
31	30	simprd 478	. . . . 5 ⊢ (𝑉 UHGrph 𝐹 → 𝐹 ∈ V)
32	9, 31	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
33		unexg 6857	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸 ∪ 𝐹) ∈ V)
34	29, 32, 33	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V)
35		isuhgra 25827	. . 3 ⊢ ((𝑉 ∈ dom UHGrph ∧ (𝐸 ∪ 𝐹) ∈ V) → (𝑉 UHGrph (𝐸 ∪ 𝐹) ↔ (𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
36	26, 34, 35	syl2anc 691	. 2 ⊢ (𝜑 → (𝑉 UHGrph (𝐸 ∪ 𝐹) ↔ (𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
37	23, 36	mpbird 246	1 ⊢ (𝜑 → 𝑉 UHGrph (𝐸 ∪ 𝐹))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800   UHGrph cuhg 25819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-uhgra 25821

This theorem is referenced by: (None)