Theorem usgrunop 15992

Description: The union of two simple graphs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple graphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a multigraph (not necessarily a simple graph!) - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
usgrunop	⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UMGraph)

Proof of Theorem usgrunop

Step	Hyp	Ref	Expression
1		usgrun.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USGraph)
2		usgrumgr 15982	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝐺 ∈ UMGraph)
4		usgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USGraph)
5		usgrumgr 15982	. . 3 ⊢ (𝐻 ∈ USGraph → 𝐻 ∈ UMGraph)
6	4, 5	syl 14	. 2 ⊢ (𝜑 → 𝐻 ∈ UMGraph)
7		usgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
8		usgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
9		usgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
10		usgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
11		usgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
12	3, 6, 7, 8, 9, 10, 11	umgrunop 15927	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UMGraph)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200   ∪ cun 3195   ∩ cin 3196  ∅c0 3491  ⟨cop 3669  dom cdm 4719  ‘cfv 5318  Vtxcvtx 15813  iEdgciedg 15814  UMGraphcumgr 15892  USGraphcusgr 15952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-sub 8319  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-dec 9579  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-umgren 15894  df-usgren 15954

This theorem is referenced by: (None)