Theorem uhgrissubgr 27059

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 2823	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 27058	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7		3simpa 1144	. . 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 769	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝑉 ⊆ 𝐴)
10		simp2 1133	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → Fun 𝐵)
11		simpr 487	. . . . . 6 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵) → 𝐼 ⊆ 𝐵)
12		funssres 6400	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
13	10, 11, 12	syl2an 597	. . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
14	13	eqcomd 2829	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15		edguhgr 26916	. . . . . . . . 9 ⊢ ((𝑆 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝑆)) → 𝑒 ∈ 𝒫 (Vtx‘𝑆))
16	15	ex 415	. . . . . . . 8 ⊢ (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
17	1	pweqi 4559	. . . . . . . . 9 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
18	17	eleq2i 2906	. . . . . . . 8 ⊢ (𝑒 ∈ 𝒫 𝑉 ↔ 𝑒 ∈ 𝒫 (Vtx‘𝑆))
19	16, 18	syl6ibr 254	. . . . . . 7 ⊢ (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
20	19	ssrdv 3975	. . . . . 6 ⊢ (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21	20	3ad2ant3 1131	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
22	21	adantr 483	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
23	1, 2, 3, 4, 5	issubgr 27055	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24	23	3adant2 1127	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
25	24	adantr 483	. . . 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 1338	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) ∧ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)) → 𝑆 SubGraph 𝐺)
27	26	ex 415	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → ((𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵) → 𝑆 SubGraph 𝐺))
28	8, 27	impbid2 228	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  𝒫 cpw 4541   class class class wbr 5068  dom cdm 5557   ↾ cres 5559  Fun wfun 6351  ‘cfv 6357  Vtxcvtx 26783  iEdgciedg 26784  Edgcedg 26834  UHGraphcuhgr 26843   SubGraph csubgr 27051

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-edg 26835  df-uhgr 26845  df-subgr 27052

This theorem is referenced by:  uhgrsubgrself  27064