Theorem subgruhgredgd 29634

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 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4		subgruhgredgd.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
5		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2763	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
7	2, 3, 4, 5, 6	subgrprop2 29624	. . 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 29632	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
12	10, 1, 11	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → Fun (iEdg‘𝑆))
13		subgruhgredgd.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ dom 𝐼)
14	4	dmeqi 5894	. . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
15	13, 14	eleqtrdi 2873	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝑆))
16	12, 15	jca 520	. . . . . . 7 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
17	16	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
18	4	fveq1i 6882	. . . . . . 7 ⊢ (𝐼‘𝑋) = ((iEdg‘𝑆)‘𝑋)
19		fvelrn 7071	. . . . . . 7 ⊢ ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
20	18, 19	eqeltrid 2867	. . . . . 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 29399	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
23	21, 22	eleqtrrdi 2874	. . . 4 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (Edg‘𝑆))
24	9, 23	sseldd 3938	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ 𝒫 𝑉)
25	5	uhgrfun 29416	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
26	10, 25	syl 18	. . . . . 6 ⊢ (𝜑 → Fun (iEdg‘𝐺))
27	26	adantr 485	. . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
30		funssfv 6902	. . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
31	30	eqcomd 2769	. . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph)
34	26	funfnd 6567	. . . . . 6 ⊢ (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
35	34	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
36		subgreldmiedg 29633	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
37	1, 15, 36	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ dom (iEdg‘𝐺))
38	37	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
39	5	uhgrn0 29417	. . . . 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 3025	. . 3 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ≠ ∅)
42		eldifsn 4753	. . 3 ⊢ ((𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ (𝐼‘𝑋) ≠ ∅))
43	24, 41, 42	sylanbrc 594	. 2 ⊢ ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
44	8, 43	mpdan 699	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589   class class class wbr 5109  dom cdm 5661  ran crn 5662  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  Vtxcvtx 29346  iEdgciedg 29347  Edgcedg 29397  UHGraphcuhgr 29406   SubGraph csubgr 29617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-edg 29398  df-uhgr 29408  df-subgr 29618

This theorem is referenced by:  subumgredg2  29635  subuhgr  29636  subupgr  29637