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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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