Theorem uhgrissubgr 27545

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 2738	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 27544	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7		3simpa 1146	. . 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 767	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝑉 ⊆ 𝐴)
10		simp2 1135	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → Fun 𝐵)
11		simpr 484	. . . . . 6 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵) → 𝐼 ⊆ 𝐵)
12		funssres 6462	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
13	10, 11, 12	syl2an 595	. . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
14	13	eqcomd 2744	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15		edguhgr 27402	. . . . . . . . 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 4548	. . . . . . . . 9 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
18	17	eleq2i 2830	. . . . . . . 8 ⊢ (𝑒 ∈ 𝒫 𝑉 ↔ 𝑒 ∈ 𝒫 (Vtx‘𝑆))
19	16, 18	syl6ibr 251	. . . . . . 7 ⊢ (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
20	19	ssrdv 3923	. . . . . 6 ⊢ (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21	20	3ad2ant3 1133	. . . . 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 27541	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24	23	3adant2 1129	. . . . 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 1340	. . 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 225	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  dom cdm 5580   ↾ cres 5582  Fun wfun 6412  ‘cfv 6418  Vtxcvtx 27269  iEdgciedg 27270  Edgcedg 27320  UHGraphcuhgr 27329   SubGraph csubgr 27537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-edg 27321  df-uhgr 27331  df-subgr 27538

This theorem is referenced by:  uhgrsubgrself  27550