Theorem upgrunop 29630

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 5431	. . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
10	5	fvexi 6887	. . . 4 ⊢ 𝑉 ∈ V
11	3	fvexi 6887	. . . . 5 ⊢ 𝐸 ∈ V
12	4	fvexi 6887	. . . . 5 ⊢ 𝐹 ∈ V
13	11, 12	unex 7744	. . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V
14	10, 13	pm3.2i 476	. . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V)
15		opvtxfv 29515	. . 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 29518	. . 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 29629	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∪ cun 3896   ∩ cin 3897  ∅c0 4278  ⟨cop 4589  dom cdm 5647  ‘cfv 6527  Vtxcvtx 29507  iEdgciedg 29508  UPGraphcupgr 29591

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-1st 7984  df-2nd 7985  df-vtx 29509  df-iedg 29510  df-upgr 29593

This theorem is used by:  uspgrunop  29703