Theorem subgruhgredgd 40508

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 2610	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4		subgruhgredgd.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
5		eqid 2610	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2610	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
7	2, 3, 4, 5, 6	subgrprop2 40498	. . 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 1062	. . . 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 40506	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
12	10, 1, 11	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → Fun (iEdg‘𝑆))
13		subgruhgredgd.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ dom 𝐼)
14	4	dmeqi 5247	. . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
15	13, 14	syl6eleq 2698	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝑆))
16	12, 15	jca 553	. . . . . . 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 6104	. . . . . . 7 ⊢ (𝐼‘𝑋) = ((iEdg‘𝑆)‘𝑋)
19		fvelrn 6260	. . . . . . 7 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
20	18, 19	syl5eqel 2692	. . . . . 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		subgrv 40494	. . . . . . . . 9 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
23	22	simpld 474	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → 𝑆 ∈ V)
24	1, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
25		edgaval 25794	. . . . . . 7 ⊢ (𝑆 ∈ V → (Edg‘𝑆) = ran (iEdg‘𝑆))
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (Edg‘𝑆) = ran (iEdg‘𝑆))
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) = ran (iEdg‘𝑆))
28	21, 27	eleqtrrd 2691	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (Edg‘𝑆))
29	9, 28	sseldd 3569	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ 𝒫 𝑉)
30	5	uhgrfun 25732	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
31	10, 30	syl 17	. . . . . 6 ⊢ (𝜑 → Fun (iEdg‘𝐺))
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
33		simpr2 1061	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
34	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
35		funssfv 6119	. . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
36	35	eqcomd 2616	. . . . 5 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
37	32, 33, 34, 36	syl3anc 1318	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
38	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph )
39		funfn 5833	. . . . . . 7 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
40	31, 39	sylib 207	. . . . . 6 ⊢ (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
41	40	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
42		subgreldmiedg 40507	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
43	1, 15, 42	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝐺))
44	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
45	5	uhgrn0 25733	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
46	38, 41, 44, 45	syl3anc 1318	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
47	37, 46	eqnetrd 2849	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ≠ ∅)
48		eldifsn 4260	. . 3 ⊢ ((𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ (𝐼‘𝑋) ≠ ∅))
49	29, 47, 48	sylanbrc 695	. 2 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
50	8, 49	mpdan 699	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792   SubGraph csubgr 40491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724  df-edga 25793  df-subgr 40492

This theorem is referenced by:  subumgredg2  40509  subuhgr  40510  subupgr  40511