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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 (𝜑𝐸 Fn 𝐴)
umgraun.f (𝜑𝐹 Fn 𝐵)
umgraun.i (𝜑 → (𝐴𝐵) = ∅)
umgraun.ge (𝜑𝑉 UMGrph 𝐸)
umgraun.gf (𝜑𝑉 UMGrph 𝐹)
Assertion
Ref Expression
umgraun (𝜑𝑉 UMGrph (𝐸𝐹))

Proof of Theorem umgraun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 umgraun.ge . . . . 5 (𝜑𝑉 UMGrph 𝐸)
2 umgraun.e . . . . 5 (𝜑𝐸 Fn 𝐴)
3 umgraf 25847 . . . . 5 ((𝑉 UMGrph 𝐸𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
41, 2, 3syl2anc 691 . . . 4 (𝜑𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
5 umgraun.gf . . . . 5 (𝜑𝑉 UMGrph 𝐹)
6 umgraun.f . . . . 5 (𝜑𝐹 Fn 𝐵)
7 umgraf 25847 . . . . 5 ((𝑉 UMGrph 𝐹𝐹 Fn 𝐵) → 𝐹:𝐵⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
85, 6, 7syl2anc 691 . . . 4 (𝜑𝐹:𝐵⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
9 umgraun.i . . . 4 (𝜑 → (𝐴𝐵) = ∅)
10 fun2 5980 . . . 4 (((𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ 𝐹:𝐵⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) ∧ (𝐴𝐵) = ∅) → (𝐸𝐹):(𝐴𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
114, 8, 9, 10syl21anc 1317 . . 3 (𝜑 → (𝐸𝐹):(𝐴𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
12 fdm 5964 . . . . 5 ((𝐸𝐹):(𝐴𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → dom (𝐸𝐹) = (𝐴𝐵))
1311, 12syl 17 . . . 4 (𝜑 → dom (𝐸𝐹) = (𝐴𝐵))
1413feq2d 5944 . . 3 (𝜑 → ((𝐸𝐹):dom (𝐸𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝐸𝐹):(𝐴𝐵)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
1511, 14mpbird 246 . 2 (𝜑 → (𝐸𝐹):dom (𝐸𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
16 relumgra 25843 . . . 4 Rel UMGrph
17 releldm 5279 . . . 4 ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ dom UMGrph )
1816, 1, 17sylancr 694 . . 3 (𝜑𝑉 ∈ dom UMGrph )
1916brrelex2i 5083 . . . . 5 (𝑉 UMGrph 𝐸𝐸 ∈ V)
201, 19syl 17 . . . 4 (𝜑𝐸 ∈ V)
2116brrelex2i 5083 . . . . 5 (𝑉 UMGrph 𝐹𝐹 ∈ V)
225, 21syl 17 . . . 4 (𝜑𝐹 ∈ V)
23 unexg 6857 . . . 4 ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸𝐹) ∈ V)
2420, 22, 23syl2anc 691 . . 3 (𝜑 → (𝐸𝐹) ∈ V)
25 isumgra 25844 . . 3 ((𝑉 ∈ dom UMGrph ∧ (𝐸𝐹) ∈ V) → (𝑉 UMGrph (𝐸𝐹) ↔ (𝐸𝐹):dom (𝐸𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
2618, 24, 25syl2anc 691 . 2 (𝜑 → (𝑉 UMGrph (𝐸𝐹) ↔ (𝐸𝐹):dom (𝐸𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
2715, 26mpbird 246 1 (𝜑𝑉 UMGrph (𝐸𝐹))
 Colors of variables: wff setvar class 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
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