Theorem uhgrun 16068

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 16055	. . . . 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 2232	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8		uhgrun.f	. . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻)
9	7, 8	uhgrfm 16055	. . . . . 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 2238	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻))
13	12	pweqd 3673	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
14	13	rabeqdv 2806	. . . . . 6 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 (Vtx‘𝐻) ∣ ∃𝑗 𝑗 ∈ 𝑠})
15	14	feq3d 5496	. . . . 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 5537	. . 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 4957	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹))
21		dmun 4962	. . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹)
22	20, 21	eqtrdi 2281	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23		uhgrun.v	. . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
24	23	pweqd 3673	. . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
25	24	rabeqdv 2806	. . . 4 ⊢ (𝜑 → {𝑠 ∈ 𝒫 (Vtx‘𝑈) ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
26	19, 22, 25	feq123d 5498	. . 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 2232	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
30		eqid 2232	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
31	29, 30	isuhgrm 16053	. . 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 2203  {crab 2524   ∪ cun 3208   ∩ cin 3209  ∅c0 3507  𝒫 cpw 3668  dom cdm 4748  ⟶wf 5347  ‘cfv 5351  Vtxcvtx 15994  iEdgciedg 15995  UHGraphcuhgr 16049

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-uhgrm 16051

This theorem is referenced by:  uhgrunop  16069  ushgrun  16070