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Theorem uhgrissubgr 40499
 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 2610 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 40498 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7 3simpa 1051 . . 3 ((𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉) → (𝑉𝐴𝐼𝐵))
86, 7syl 17 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵))
9 simprl 790 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) ∧ (𝑉𝐴𝐼𝐵)) → 𝑉𝐴)
10 simp2 1055 . . . . . 6 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) → Fun 𝐵)
11 simpr 476 . . . . . 6 ((𝑉𝐴𝐼𝐵) → 𝐼𝐵)
12 funssres 5844 . . . . . 6 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1310, 11, 12syl2an 493 . . . . 5 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) ∧ (𝑉𝐴𝐼𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
1413eqcomd 2616 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) ∧ (𝑉𝐴𝐼𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15 edguhgr 25803 . . . . . . . . 9 ((𝑆 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝑆)) → 𝑒 ∈ 𝒫 (Vtx‘𝑆))
1615ex 449 . . . . . . . 8 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
171pweqi 4112 . . . . . . . . 9 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
1817eleq2i 2680 . . . . . . . 8 (𝑒 ∈ 𝒫 𝑉𝑒 ∈ 𝒫 (Vtx‘𝑆))
1916, 18syl6ibr 241 . . . . . . 7 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
2019ssrdv 3574 . . . . . 6 (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21203ad2ant3 1077 . . . . 5 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
2221adantr 480 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) ∧ (𝑉𝐴𝐼𝐵)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
231, 2, 3, 4, 5issubgr 40495 . . . . . 6 ((𝐺𝑊𝑆 ∈ UHGraph ) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24233adant2 1073 . . . . 5 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
2524adantr 480 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) ∧ (𝑉𝐴𝐼𝐵)) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
269, 14, 22, 25mpbir3and 1238 . . 3 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) ∧ (𝑉𝐴𝐼𝐵)) → 𝑆 SubGraph 𝐺)
2726ex 449 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) → ((𝑉𝐴𝐼𝐵) → 𝑆 SubGraph 𝐺))
288, 27impbid2 215 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph ) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  Fun wfun 5798  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792   SubGraph csubgr 40491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724  df-edga 25793  df-subgr 40492 This theorem is referenced by:  uhgrsubgrself  40504
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