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Theorem uhgrissubgr 26509
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 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))

Proof of Theorem uhgrissubgr
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef 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 2799 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 26508 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7 3simpa 1179 . . 3 ((𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉) → (𝑉𝐴𝐼𝐵))
86, 7syl 17 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵))
9 simprl 788 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝑉𝐴)
10 simp2 1168 . . . . . 6 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → Fun 𝐵)
11 simpr 478 . . . . . 6 ((𝑉𝐴𝐼𝐵) → 𝐼𝐵)
12 funssres 6144 . . . . . 6 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1310, 11, 12syl2an 590 . . . . 5 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
1413eqcomd 2805 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15 edguhgr 26364 . . . . . . . . 9 ((𝑆 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝑆)) → 𝑒 ∈ 𝒫 (Vtx‘𝑆))
1615ex 402 . . . . . . . 8 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
171pweqi 4353 . . . . . . . . 9 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
1817eleq2i 2870 . . . . . . . 8 (𝑒 ∈ 𝒫 𝑉𝑒 ∈ 𝒫 (Vtx‘𝑆))
1916, 18syl6ibr 244 . . . . . . 7 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
2019ssrdv 3804 . . . . . 6 (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21203ad2ant3 1166 . . . . 5 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
2221adantr 473 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
231, 2, 3, 4, 5issubgr 26505 . . . . . 6 ((𝐺𝑊𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24233adant2 1162 . . . . 5 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
2524adantr 473 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
269, 14, 22, 25mpbir3and 1443 . . 3 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝑆 SubGraph 𝐺)
2726ex 402 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → ((𝑉𝐴𝐼𝐵) → 𝑆 SubGraph 𝐺))
288, 27impbid2 218 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wss 3769  𝒫 cpw 4349   class class class wbr 4843  dom cdm 5312  cres 5314  Fun wfun 6095  cfv 6101  Vtxcvtx 26231  iEdgciedg 26232  Edgcedg 26282  UHGraphcuhgr 26291   SubGraph csubgr 26501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-edg 26283  df-uhgr 26293  df-subgr 26502
This theorem is referenced by:  uhgrsubgrself  26514
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