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Theorem uhgrun 26190
Description: The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
uhgrun.g (𝜑𝐺 ∈ UHGraph)
uhgrun.h (𝜑𝐻 ∈ UHGraph)
uhgrun.e 𝐸 = (iEdg‘𝐺)
uhgrun.f 𝐹 = (iEdg‘𝐻)
uhgrun.vg 𝑉 = (Vtx‘𝐺)
uhgrun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
uhgrun.i (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
uhgrun.u (𝜑𝑈𝑊)
uhgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
uhgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
uhgrun (𝜑𝑈 ∈ UHGraph)

Proof of Theorem uhgrun
StepHypRef Expression
1 uhgrun.g . . . . 5 (𝜑𝐺 ∈ UHGraph)
2 uhgrun.vg . . . . . 6 𝑉 = (Vtx‘𝐺)
3 uhgrun.e . . . . . 6 𝐸 = (iEdg‘𝐺)
42, 3uhgrf 26178 . . . . 5 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
51, 4syl 17 . . . 4 (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
6 uhgrun.h . . . . . 6 (𝜑𝐻 ∈ UHGraph)
7 eqid 2771 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 uhgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8uhgrf 26178 . . . . . 6 (𝐻 ∈ UHGraph → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
106, 9syl 17 . . . . 5 (𝜑𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
11 uhgrun.vh . . . . . . . . 9 (𝜑 → (Vtx‘𝐻) = 𝑉)
1211eqcomd 2777 . . . . . . . 8 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4303 . . . . . . 7 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 3878 . . . . . 6 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514feq3d 6171 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})))
1610, 15mpbird 247 . . . 4 (𝜑𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
17 uhgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
185, 16, 17fun2d 6209 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
19 uhgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2019dmeqd 5463 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
21 dmun 5468 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2220, 21syl6eq 2821 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23 uhgrun.v . . . . . 6 (𝜑 → (Vtx‘𝑈) = 𝑉)
2423pweqd 4303 . . . . 5 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2524difeq1d 3878 . . . 4 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2619, 22, 25feq123d 6173 . . 3 (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
2718, 26mpbird 247 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))
28 uhgrun.u . . 3 (𝜑𝑈𝑊)
29 eqid 2771 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
30 eqid 2771 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3129, 30isuhgr 26176 . . 3 (𝑈𝑊 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
3228, 31syl 17 . 2 (𝜑 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
3327, 32mpbird 247 1 (𝜑𝑈 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  cdif 3720  cun 3721  cin 3722  c0 4063  𝒫 cpw 4298  {csn 4317  dom cdm 5250  wf 6026  cfv 6030  Vtxcvtx 26095  iEdgciedg 26096  UHGraphcuhgr 26172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-uhgr 26174
This theorem is referenced by:  uhgrunop  26191  ushgrun  26192
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