Theorem uhgrspansubgrlem 29325

Description: Lemma for uhgrspansubgr 29326: 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 29326. (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 29084	. . . 4 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
2	1	eleq2i 2836	. . 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 29101	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
6		funres 6620	. . . . . . 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 6600	. . . . . 6 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸 ↾ 𝐴)))
10	7, 9	mpbird 257	. . . . 5 ⊢ (𝜑 → Fun (iEdg‘𝑆))
11		elrnrexdmb 7124	. . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
14	13	fveq1d 6922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸 ↾ 𝐴)‘𝑖))
15	8	dmeqd 5930	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐴))
16		dmres 6041	. . . . . . . . . . . . 13 ⊢ dom (𝐸 ↾ 𝐴) = (𝐴 ∩ dom 𝐸)
17	15, 16	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
18	17	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19		elinel1 4224	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ 𝐴)
20	18, 19	biimtrdi 253	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ 𝐴))
21	20	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ 𝐴)
22	21	fvresd 6940	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸 ↾ 𝐴)‘𝑖) = (𝐸‘𝑖))
23	14, 22	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸‘𝑖))
24		elinel2 4225	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
25	18, 24	biimtrdi 253	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
26	25	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27		uhgrspan.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
28	27, 4	uhgrss 29099	. . . . . . . . 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 4639	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
32	31	eleq2d 2830	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
34		fvex 6933	. . . . . . . . . 10 ⊢ (𝐸‘𝑖) ∈ V
35	34	elpw 4626	. . . . . . . . 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 2844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39		eleq1 2832	. . . . . 6 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
40	38, 39	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
41	40	rexlimdva 3161	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
42	12, 41	sylbid 240	. . 3 ⊢ (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
43	2, 42	biimtrid 242	. 2 ⊢ (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
44	43	ssrdv 4014	1 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  dom cdm 5700  ran crn 5701   ↾ cres 5702  Fun wfun 6567  ‘cfv 6573  Vtxcvtx 29031  iEdgciedg 29032  Edgcedg 29082  UHGraphcuhgr 29091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-edg 29083  df-uhgr 29093

This theorem is referenced by:  uhgrspansubgr  29326