Theorem uhgrspansubgrlem 29700

Description: Lemma for uhgrspansubgr 29701: The edges of the graph 𝑆 obtained by removing some edges of a hypergraph 𝐺 are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 29701. (Contributed by AV, 18-Nov-2020.)

Hypotheses

Ref	Expression
uhgrspan.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrspan.e	⊢ 𝐸 = (iEdg‘𝐺)
uhgrspan.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
uhgrspan.q	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
uhgrspan.r	⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
uhgrspan.g	⊢ (𝜑 → 𝐺 ∈ UHGraph)

Assertion

Ref	Expression
uhgrspansubgrlem	⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Proof of Theorem uhgrspansubgrlem

Dummy variables 𝑒 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		edgval 29456	. . . 4 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
2	1	eleq2i 2857	. . 3 ⊢ (𝑒 ∈ (Edg‘𝑆) ↔ 𝑒 ∈ ran (iEdg‘𝑆))
3		uhgrspan.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
4		uhgrspan.e	. . . . . . . 8 ⊢ 𝐸 = (iEdg‘𝐺)
5	4	uhgrfun 29473	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
6		funres 6582	. . . . . . 7 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐴))
7	3, 5, 6	3syl 19	. . . . . 6 ⊢ (𝜑 → Fun (𝐸 ↾ 𝐴))
8		uhgrspan.r	. . . . . . 7 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
9	8	funeqd 6562	. . . . . 6 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸 ↾ 𝐴)))
10	7, 9	mpbird 260	. . . . 5 ⊢ (𝜑 → Fun (iEdg‘𝑆))
11		elrnrexdmb 7089	. . . . 5 ⊢ (Fun (iEdg‘𝑆) → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
13	8	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
14	13	fveq1d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸 ↾ 𝐴)‘𝑖))
15	8	dmeqd 5897	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐴))
16		dmres 6013	. . . . . . . . . . . . 13 ⊢ dom (𝐸 ↾ 𝐴) = (𝐴 ∩ dom 𝐸)
17	15, 16	eqtrdi 2816	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
18	17	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19		elinel1 4154	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ 𝐴)
20	18, 19	biimtrdi 256	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ 𝐴))
21	20	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ 𝐴)
22	21	fvresd 6905	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸 ↾ 𝐴)‘𝑖) = (𝐸‘𝑖))
23	14, 22	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸‘𝑖))
24		elinel2 4155	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
25	18, 24	biimtrdi 256	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
26	25	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27		uhgrspan.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
28	27, 4	uhgrss 29471	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ⊆ 𝑉)
29	3, 26, 28	syl2an2r 698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸‘𝑖) ⊆ 𝑉)
30		uhgrspan.q	. . . . . . . . . . . 12 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
31	30	pweqd 4581	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
32	31	eleq2d 2851	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
33	32	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
34		fvex 6898	. . . . . . . . . 10 ⊢ (𝐸‘𝑖) ∈ V
35	34	elpw 4568	. . . . . . . . 9 ⊢ ((𝐸‘𝑖) ∈ 𝒫 𝑉 ↔ (𝐸‘𝑖) ⊆ 𝑉)
36	33, 35	bitrdi 290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ⊆ 𝑉))
37	29, 36	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
38	23, 37	eqeltrd 2865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39		eleq1 2853	. . . . . 6 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
40	38, 39	syl5ibrcom 250	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
41	40	rexlimdva 3168	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
42	12, 41	sylbid 243	. . 3 ⊢ (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
43	2, 42	biimtrid 245	. 2 ⊢ (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
44	43	ssrdv 3944	1 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564  dom cdm 5663  ran crn 5664   ↾ cres 5665  Fun wfun 6534  ‘cfv 6540  Vtxcvtx 29403  iEdgciedg 29404  Edgcedg 29454  UHGraphcuhgr 29463

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-edg 29455  df-uhgr 29465

This theorem is used by:  uhgrspansubgr  29701