Theorem subgruhgredgd 29084

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 2727	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4		subgruhgredgd.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
5		eqid 2727	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2727	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
7	2, 3, 4, 5, 6	subgrprop2 29074	. . 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 1194	. . . 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 29082	. . . . . . . . 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 5901	. . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
15	13, 14	eleqtrdi 2838	. . . . . . . 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 6892	. . . . . . 7 ⊢ (𝐼‘𝑋) = ((iEdg‘𝑆)‘𝑋)
19		fvelrn 7080	. . . . . . 7 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
20	18, 19	eqeltrid 2832	. . . . . 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 28849	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
23	21, 22	eleqtrrdi 2839	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (Edg‘𝑆))
24	9, 23	sseldd 3979	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ 𝒫 𝑉)
25	5	uhgrfun 28866	. . . . . . 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 1193	. . . . 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 6912	. . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
31	30	eqcomd 2733	. . . . 5 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
32	27, 28, 29, 31	syl3anc 1369	. . . 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 6578	. . . . . 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 29083	. . . . . . 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 28867	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
40	33, 35, 38, 39	syl3anc 1369	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
41	32, 40	eqnetrd 3003	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ≠ ∅)
42		eldifsn 4786	. . 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 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935   ∖ cdif 3941   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598  {csn 4624   class class class wbr 5142  dom cdm 5672  ran crn 5673  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542  Vtxcvtx 28796  iEdgciedg 28797  Edgcedg 28847  UHGraphcuhgr 28856   SubGraph csubgr 29067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-edg 28848  df-uhgr 28858  df-subgr 29068

This theorem is referenced by:  subumgredg2  29085  subuhgr  29086  subupgr  29087