Theorem subgruhgredgd 29319

Description: An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)

Hypotheses

Ref	Expression
subgruhgredgd.v	⊢ 𝑉 = (Vtx‘𝑆)
subgruhgredgd.i	⊢ 𝐼 = (iEdg‘𝑆)
subgruhgredgd.g	⊢ (𝜑 → 𝐺 ∈ UHGraph)
subgruhgredgd.s	⊢ (𝜑 → 𝑆 SubGraph 𝐺)
subgruhgredgd.x	⊢ (𝜑 → 𝑋 ∈ dom 𝐼)

Assertion

Ref	Expression
subgruhgredgd	⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Proof of Theorem subgruhgredgd

Step	Hyp	Ref	Expression
1		subgruhgredgd.s	. . 3 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)
2		subgruhgredgd.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
3		eqid 2740	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4		subgruhgredgd.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
5		eqid 2740	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2740	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
7	2, 3, 4, 5, 6	subgrprop2 29309	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
8	1, 7	syl 17	. 2 ⊢ (𝜑 → (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
9		simpr3 1196	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
10		subgruhgredgd.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
11		subgruhgrfun 29317	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
12	10, 1, 11	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → Fun (iEdg‘𝑆))
13		subgruhgredgd.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ dom 𝐼)
14	4	dmeqi 5929	. . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
15	13, 14	eleqtrdi 2854	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝑆))
16	12, 15	jca 511	. . . . . . 7 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
18	4	fveq1i 6921	. . . . . . 7 ⊢ (𝐼‘𝑋) = ((iEdg‘𝑆)‘𝑋)
19		fvelrn 7110	. . . . . . 7 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
20	18, 19	eqeltrid 2848	. . . . . 6 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → (𝐼‘𝑋) ∈ ran (iEdg‘𝑆))
21	17, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ ran (iEdg‘𝑆))
22		edgval 29084	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
23	21, 22	eleqtrrdi 2855	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (Edg‘𝑆))
24	9, 23	sseldd 4009	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ 𝒫 𝑉)
25	5	uhgrfun 29101	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
26	10, 25	syl 17	. . . . . 6 ⊢ (𝜑 → Fun (iEdg‘𝐺))
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
28		simpr2 1195	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
29	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
30		funssfv 6941	. . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
31	30	eqcomd 2746	. . . . 5 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
32	27, 28, 29, 31	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
33	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph)
34	26	funfnd 6609	. . . . . 6 ⊢ (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
35	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
36		subgreldmiedg 29318	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
37	1, 15, 36	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝐺))
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
39	5	uhgrn0 29102	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
40	33, 35, 38, 39	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
41	32, 40	eqnetrd 3014	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ≠ ∅)
42		eldifsn 4811	. . 3 ⊢ ((𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ (𝐼‘𝑋) ≠ ∅))
43	24, 41, 42	sylanbrc 582	. 2 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
44	8, 43	mpdan 686	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166  dom cdm 5700  ran crn 5701  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  Vtxcvtx 29031  iEdgciedg 29032  Edgcedg 29082  UHGraphcuhgr 29091   SubGraph csubgr 29302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-edg 29083  df-uhgr 29093  df-subgr 29303

This theorem is referenced by:  subumgredg2  29320  subuhgr  29321  subupgr  29322