Theorem uhgrspansubgrlem 29491

Description: Lemma for uhgrspansubgr 29492: 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 29492. (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 29250	. . . 4 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
2	1	eleq2i 2854	. . 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 29267	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
6		funres 6563	. . . . . . 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 6543	. . . . . 6 ⊢ (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸 ↾ 𝐴)))
10	7, 9	mpbird 259	. . . . 5 ⊢ (𝜑 → Fun (iEdg‘𝑆))
11		elrnrexdmb 7071	. . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
14	13	fveq1d 6869	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸 ↾ 𝐴)‘𝑖))
15	8	dmeqd 5881	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐴))
16		dmres 5998	. . . . . . . . . . . . 13 ⊢ dom (𝐸 ↾ 𝐴) = (𝐴 ∩ dom 𝐸)
17	15, 16	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
18	17	eleq2d 2848	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19		elinel1 4153	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ 𝐴)
20	18, 19	biimtrdi 255	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ 𝐴))
21	20	imp 410	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ 𝐴)
22	21	fvresd 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸 ↾ 𝐴)‘𝑖) = (𝐸‘𝑖))
23	14, 22	eqtrd 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸‘𝑖))
24		elinel2 4154	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
25	18, 24	biimtrdi 255	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
26	25	imp 410	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27		uhgrspan.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
28	27, 4	uhgrss 29265	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ⊆ 𝑉)
29	3, 26, 28	syl2an2r 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸‘𝑖) ⊆ 𝑉)
30		uhgrspan.q	. . . . . . . . . . . 12 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
31	30	pweqd 4572	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
32	31	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
33	32	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ∈ 𝒫 𝑉))
34		fvex 6880	. . . . . . . . . 10 ⊢ (𝐸‘𝑖) ∈ V
35	34	elpw 4559	. . . . . . . . 9 ⊢ ((𝐸‘𝑖) ∈ 𝒫 𝑉 ↔ (𝐸‘𝑖) ⊆ 𝑉)
36	33, 35	bitrdi 289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸‘𝑖) ⊆ 𝑉))
37	29, 36	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
38	23, 37	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39		eleq1 2850	. . . . . 6 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
40	38, 39	syl5ibrcom 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
41	40	rexlimdva 3163	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
42	12, 41	sylbid 242	. . 3 ⊢ (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
43	2, 42	biimtrid 244	. 2 ⊢ (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
44	43	ssrdv 3942	1 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4555  dom cdm 5647  ran crn 5648   ↾ cres 5649  Fun wfun 6515  ‘cfv 6521  Vtxcvtx 29197  iEdgciedg 29198  Edgcedg 29248  UHGraphcuhgr 29257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-edg 29249  df-uhgr 29259

This theorem is referenced by:  uhgrspansubgr  29492