Theorem subgruhgredgd 29745

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 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4		subgruhgredgd.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
5		eqid 2760	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2760	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
7	2, 3, 4, 5, 6	subgrprop2 29735	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
8	1, 7	syl 18	. 2 ⊢ (𝜑 → (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
9		simpr3 1215	. . . 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 29743	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
12	10, 1, 11	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → Fun (iEdg‘𝑆))
13		subgruhgredgd.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ dom 𝐼)
14	4	dmeqi 5888	. . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
15	13, 14	eleqtrdi 2870	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝑆))
16	12, 15	jca 521	. . . . . . 7 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
17	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
18	4	fveq1i 6880	. . . . . . 7 ⊢ (𝐼‘𝑋) = ((iEdg‘𝑆)‘𝑋)
19		fvelrn 7070	. . . . . . 7 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
20	18, 19	eqeltrid 2864	. . . . . 6 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → (𝐼‘𝑋) ∈ ran (iEdg‘𝑆))
21	17, 20	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ ran (iEdg‘𝑆))
22		edgval 29507	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
23	21, 22	eleqtrrdi 2871	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (Edg‘𝑆))
24	9, 23	sseldd 3932	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ 𝒫 𝑉)
25	5	uhgrfun 29524	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
26	10, 25	syl 18	. . . . . 6 ⊢ (𝜑 → Fun (iEdg‘𝐺))
27	26	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
28		simpr2 1214	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
29	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
30		funssfv 6900	. . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
31	30	eqcomd 2766	. . . . 5 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
32	27, 28, 29, 31	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
33	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph)
34	26	funfnd 6565	. . . . . 6 ⊢ (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
35	34	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
36		subgreldmiedg 29744	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
37	1, 15, 36	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝐺))
38	37	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
39	5	uhgrn0 29525	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
40	33, 35, 38, 39	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
41	32, 40	eqnetrd 3022	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ≠ ∅)
42		eldifsn 4748	. . 3 ⊢ ((𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ (𝐼‘𝑋) ≠ ∅))
43	24, 41, 42	sylanbrc 595	. 2 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
44	8, 43	mpdan 700	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103  dom cdm 5655  ran crn 5656  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533  Vtxcvtx 29454  iEdgciedg 29455  Edgcedg 29505  UHGraphcuhgr 29514   SubGraph csubgr 29728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-edg 29506  df-uhgr 29516  df-subgr 29729

This theorem is used by:  subumgredg2  29746  subuhgr  29747  subupgr  29748