Theorem uhgrun 16210

Description: The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex set 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
uhgrun.g	⊢ (𝜑 → 𝐺 ∈ UHGraph)
uhgrun.h	⊢ (𝜑 → 𝐻 ∈ 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	⊢ (𝜑 → 𝑈 ∈ 𝑊)
uhgrun.v	⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
uhgrun.un	⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))

Assertion

Ref	Expression
uhgrun	⊢ (𝜑 → 𝑈 ∈ UHGraph)

Proof of Theorem uhgrun

Dummy variables 𝑠 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrun.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
2		uhgrun.vg	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3		uhgrun.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	uhgrfm 16197	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
5	1, 4	syl 14	. . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
6		uhgrun.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ UHGraph)
7		eqid 2234	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		uhgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	uhgrfm 16197	. . . . . 6 ⊢ (𝐻 ∈ UHGraph → 𝐹:dom 𝐹⟶{𝑠 ∈ 𝒫 (Vtx‘𝐻) ∣ ∃𝑗 𝑗 ∈ 𝑠})
10	6, 9	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑠 ∈ 𝒫 (Vtx‘𝐻) ∣ ∃𝑗 𝑗 ∈ 𝑠})
11		uhgrun.vh	. . . . . . . . 9 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
12	11	eqcomd 2240	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 3679	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	rabeqdv 2809	. . . . . 6 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 (Vtx‘𝐻) ∣ ∃𝑗 𝑗 ∈ 𝑠})
15	14	feq3d 5502	. . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ 𝐹:dom 𝐹⟶{𝑠 ∈ 𝒫 (Vtx‘𝐻) ∣ ∃𝑗 𝑗 ∈ 𝑠}))
16	10, 15	mpbird 167	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
17		uhgrun.i	. . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
18	5, 16, 17	fun2d 5543	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
19		uhgrun.un	. . . 4 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
20	19	dmeqd 4963	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
21		dmun 4968	. . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹)
22	20, 21	eqtrdi 2283	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23		uhgrun.v	. . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
24	23	pweqd 3679	. . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
25	24	rabeqdv 2809	. . . 4 ⊢ (𝜑 → {𝑠 ∈ 𝒫 (Vtx‘𝑈) ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
26	19, 22, 25	feq123d 5504	. . 3 ⊢ (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑠 ∈ 𝒫 (Vtx‘𝑈) ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
27	18, 26	mpbird 167	. 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑠 ∈ 𝒫 (Vtx‘𝑈) ∣ ∃𝑗 𝑗 ∈ 𝑠})
28		uhgrun.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
29		eqid 2234	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
30		eqid 2234	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
31	29, 30	isuhgrm 16195	. . 3 ⊢ (𝑈 ∈ 𝑊 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑠 ∈ 𝒫 (Vtx‘𝑈) ∣ ∃𝑗 𝑗 ∈ 𝑠}))
32	28, 31	syl 14	. 2 ⊢ (𝜑 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑠 ∈ 𝒫 (Vtx‘𝑈) ∣ ∃𝑗 𝑗 ∈ 𝑠}))
33	27, 32	mpbird 167	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {crab 2526   ∪ cun 3212   ∩ cin 3213  ∅c0 3512  𝒫 cpw 3674  dom cdm 4754  ⟶wf 5353  ‘cfv 5357  Vtxcvtx 16136  iEdgciedg 16137  UHGraphcuhgr 16191

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-uhgrm 16193

This theorem is referenced by:  uhgrunop  16211  ushgrun  16212