Theorem upgrunop 29724

Description: The union of two pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
upgrun.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
upgrun.h	⊢ (𝜑 → 𝐻 ∈ UPGraph)
upgrun.e	⊢ 𝐸 = (iEdg‘𝐺)
upgrun.f	⊢ 𝐹 = (iEdg‘𝐻)
upgrun.vg	⊢ 𝑉 = (Vtx‘𝐺)
upgrun.vh	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
upgrun.i	⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)

Assertion

Ref	Expression
upgrunop	⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Proof of Theorem upgrunop

Step	Hyp	Ref	Expression
1		upgrun.g	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
2		upgrun.h	. 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
3		upgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
4		upgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
5		upgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
6		upgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
7		upgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
8		opex 5436	. . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
10	5	fvexi 6919	. . . 4 ⊢ 𝑉 ∈ V
11	3	fvexi 6919	. . . . 5 ⊢ 𝐸 ∈ V
12	4	fvexi 6919	. . . . 5 ⊢ 𝐹 ∈ V
13	11, 12	unex 7782	. . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V
14	10, 13	pm3.2i 476	. . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V)
15		opvtxfv 29609	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
16	14, 15	mp1i 14	. 2 ⊢ (𝜑 → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
17		opiedgfv 29612	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
18	14, 17	mp1i 14	. 2 ⊢ (𝜑 → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
19	1, 2, 3, 4, 5, 6, 7, 9, 16, 18	upgrun 29723	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ⟨cop 4838  dom cdm 5655  ‘cfv 6555  Vtxcvtx 29601  iEdgciedg 29602  UPGraphcupgr 29685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-1st 8022  df-2nd 8023  df-vtx 29603  df-iedg 29604  df-upgr 29687

This theorem is used by:  uspgrunop  29797