Theorem uhgrissubgr 29254

Description: The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
uhgrissubgr.v	⊢ 𝑉 = (Vtx‘𝑆)
uhgrissubgr.a	⊢ 𝐴 = (Vtx‘𝐺)
uhgrissubgr.i	⊢ 𝐼 = (iEdg‘𝑆)
uhgrissubgr.b	⊢ 𝐵 = (iEdg‘𝐺)

Assertion

Ref	Expression
uhgrissubgr	⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)))

Proof of Theorem uhgrissubgr

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrissubgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
2		uhgrissubgr.a	. . . 4 ⊢ 𝐴 = (Vtx‘𝐺)
3		uhgrissubgr.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
4		uhgrissubgr.b	. . . 4 ⊢ 𝐵 = (iEdg‘𝐺)
5		eqid 2731	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29253	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7		3simpa 1148	. . 3 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵))
8	6, 7	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵))
9		simprl 770	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝑉 ⊆ 𝐴)
10		simp2 1137	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → Fun 𝐵)
11		simpr 484	. . . . . 6 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵) → 𝐼 ⊆ 𝐵)
12		funssres 6525	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
13	10, 11, 12	syl2an 596	. . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
14	13	eqcomd 2737	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15		edguhgr 29108	. . . . . . . . 9 ⊢ ((𝑆 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝑆)) → 𝑒 ∈ 𝒫 (Vtx‘𝑆))
16	15	ex 412	. . . . . . . 8 ⊢ (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
17	1	pweqi 4566	. . . . . . . . 9 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
18	17	eleq2i 2823	. . . . . . . 8 ⊢ (𝑒 ∈ 𝒫 𝑉 ↔ 𝑒 ∈ 𝒫 (Vtx‘𝑆))
19	16, 18	imbitrrdi 252	. . . . . . 7 ⊢ (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
20	19	ssrdv 3940	. . . . . 6 ⊢ (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21	20	3ad2ant3 1135	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
22	21	adantr 480	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
23	1, 2, 3, 4, 5	issubgr 29250	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24	23	3adant2 1131	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
25	24	adantr 480	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
26	9, 14, 22, 25	mpbir3and 1343	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝑆 SubGraph 𝐺)
27	26	ex 412	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → ((𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵) → 𝑆 SubGraph 𝐺))
28	8, 27	impbid2 226	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  𝒫 cpw 4550   class class class wbr 5091  dom cdm 5616   ↾ cres 5618  Fun wfun 6475  ‘cfv 6481  Vtxcvtx 28975  iEdgciedg 28976  Edgcedg 29026  UHGraphcuhgr 29035   SubGraph csubgr 29246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-edg 29027  df-uhgr 29037  df-subgr 29247

This theorem is referenced by:  uhgrsubgrself  29259  isubgrsubgr  47906