Theorem umgraun 25857

Description: The union of two (undirected) multigraphs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are graphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)

Hypotheses

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

Assertion

Ref	Expression
umgraun	⊢ (𝜑 → 𝑉 UMGrph (𝐸 ∪ 𝐹))

Proof of Theorem umgraun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgraun.ge	. . . . 5 ⊢ (𝜑 → 𝑉 UMGrph 𝐸)
2		umgraun.e	. . . . 5 ⊢ (𝜑 → 𝐸 Fn 𝐴)
3		umgraf 25847	. . . . 5 ⊢ ((𝑉 UMGrph 𝐸 ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4	1, 2, 3	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
5		umgraun.gf	. . . . 5 ⊢ (𝜑 → 𝑉 UMGrph 𝐹)
6		umgraun.f	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐵)
7		umgraf 25847	. . . . 5 ⊢ ((𝑉 UMGrph 𝐹 ∧ 𝐹 Fn 𝐵) → 𝐹:𝐵⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
8	5, 6, 7	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
9		umgraun.i	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
10		fun2 5980	. . . 4 ⊢ (((𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ 𝐹:𝐵⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
11	4, 8, 9, 10	syl21anc 1317	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
12		fdm 5964	. . . . 5 ⊢ ((𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → dom (𝐸 ∪ 𝐹) = (𝐴 ∪ 𝐵))
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → dom (𝐸 ∪ 𝐹) = (𝐴 ∪ 𝐵))
14	13	feq2d 5944	. . 3 ⊢ (𝜑 → ((𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝐸 ∪ 𝐹):(𝐴 ∪ 𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
15	11, 14	mpbird 246	. 2 ⊢ (𝜑 → (𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
16		relumgra 25843	. . . 4 ⊢ Rel UMGrph
17		releldm 5279	. . . 4 ⊢ ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ dom UMGrph )
18	16, 1, 17	sylancr 694	. . 3 ⊢ (𝜑 → 𝑉 ∈ dom UMGrph )
19	16	brrelex2i 5083	. . . . 5 ⊢ (𝑉 UMGrph 𝐸 → 𝐸 ∈ V)
20	1, 19	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
21	16	brrelex2i 5083	. . . . 5 ⊢ (𝑉 UMGrph 𝐹 → 𝐹 ∈ V)
22	5, 21	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
23		unexg 6857	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸 ∪ 𝐹) ∈ V)
24	20, 22, 23	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V)
25		isumgra 25844	. . 3 ⊢ ((𝑉 ∈ dom UMGrph ∧ (𝐸 ∪ 𝐹) ∈ V) → (𝑉 UMGrph (𝐸 ∪ 𝐹) ↔ (𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
26	18, 24, 25	syl2anc 691	. 2 ⊢ (𝜑 → (𝑉 UMGrph (𝐸 ∪ 𝐹) ↔ (𝐸 ∪ 𝐹):dom (𝐸 ∪ 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
27	15, 26	mpbird 246	1 ⊢ (𝜑 → 𝑉 UMGrph (𝐸 ∪ 𝐹))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900  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  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979   UMGrph cumg 25841

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-umgra 25842

This theorem is referenced by:  uslgraun  25903  eupap1  26503