Theorem uhgrspansubgrlem 28547

Description: Lemma for uhgrspansubgr 28548: 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 28548. (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 28309	. . . 4 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
2	1	eleq2i 2826	. . 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 28326	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
6		funres 6591	. . . . . . 7 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐴))
7	3, 5, 6	3syl 18	. . . . . 6 ⊢ (𝜑 → Fun (𝐸 ↾ 𝐴))
8		uhgrspan.r	. . . . . . 7 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
9	8	funeqd 6571	. . . . . 6 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸 ↾ 𝐴)))
10	7, 9	mpbird 257	. . . . 5 ⊢ (𝜑 → Fun (iEdg‘𝑆))
11		elrnrexdmb 7092	. . . . 5 ⊢ (Fun (iEdg‘𝑆) → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
13	8	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
14	13	fveq1d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸 ↾ 𝐴)‘𝑖))
15	8	dmeqd 5906	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐴))
16		dmres 6004	. . . . . . . . . . . . 13 ⊢ dom (𝐸 ↾ 𝐴) = (𝐴 ∩ dom 𝐸)
17	15, 16	eqtrdi 2789	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
18	17	eleq2d 2820	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19		elinel1 4196	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ 𝐴)
20	18, 19	syl6bi 253	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ 𝐴))
21	20	imp 408	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ 𝐴)
22	21	fvresd 6912	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸 ↾ 𝐴)‘𝑖) = (𝐸‘𝑖))
23	14, 22	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸‘𝑖))
24		elinel2 4197	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
25	18, 24	syl6bi 253	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
26	25	imp 408	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27		uhgrspan.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
28	27, 4	uhgrss 28324	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ⊆ 𝑉)
29	3, 26, 28	syl2an2r 684	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸‘𝑖) ⊆ 𝑉)
30		uhgrspan.q	. . . . . . . . . . . 12 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
31	30	pweqd 4620	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
32	31	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
33	32	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
34		fvex 6905	. . . . . . . . . 10 ⊢ (𝐸‘𝑖) ∈ V
35	34	elpw 4607	. . . . . . . . 9 ⊢ ((𝐸‘𝑖) ∈ 𝒫 𝑉 ↔ (𝐸‘𝑖) ⊆ 𝑉)
36	33, 35	bitrdi 287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ⊆ 𝑉))
37	29, 36	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
38	23, 37	eqeltrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39		eleq1 2822	. . . . . 6 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
40	38, 39	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
41	40	rexlimdva 3156	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
42	12, 41	sylbid 239	. . 3 ⊢ (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
43	2, 42	biimtrid 241	. 2 ⊢ (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
44	43	ssrdv 3989	1 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  dom cdm 5677  ran crn 5678   ↾ cres 5679  Fun wfun 6538  ‘cfv 6544  Vtxcvtx 28256  iEdgciedg 28257  Edgcedg 28307  UHGraphcuhgr 28316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-edg 28308  df-uhgr 28318

This theorem is referenced by:  uhgrspansubgr  28548