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Theorem uhgrissubgr 29306
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 2734 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 29305 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7 3simpa 1147 . . 3 ((𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉) → (𝑉𝐴𝐼𝐵))
86, 7syl 17 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵))
9 simprl 771 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝑉𝐴)
10 simp2 1136 . . . . . 6 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → Fun 𝐵)
11 simpr 484 . . . . . 6 ((𝑉𝐴𝐼𝐵) → 𝐼𝐵)
12 funssres 6611 . . . . . 6 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1310, 11, 12syl2an 596 . . . . 5 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
1413eqcomd 2740 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15 edguhgr 29160 . . . . . . . . 9 ((𝑆 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝑆)) → 𝑒 ∈ 𝒫 (Vtx‘𝑆))
1615ex 412 . . . . . . . 8 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
171pweqi 4620 . . . . . . . . 9 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
1817eleq2i 2830 . . . . . . . 8 (𝑒 ∈ 𝒫 𝑉𝑒 ∈ 𝒫 (Vtx‘𝑆))
1916, 18imbitrrdi 252 . . . . . . 7 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
2019ssrdv 4000 . . . . . 6 (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21203ad2ant3 1134 . . . . 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 29302 . . . . . 6 ((𝐺𝑊𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24233adant2 1130 . . . . 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 1341 . . 3 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝑆 SubGraph 𝐺)
2726ex 412 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → ((𝑉𝐴𝐼𝐵) → 𝑆 SubGraph 𝐺))
288, 27impbid2 226 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wss 3962  𝒫 cpw 4604   class class class wbr 5147  dom cdm 5688  cres 5690  Fun wfun 6556  cfv 6562  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UHGraphcuhgr 29087   SubGraph csubgr 29298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-edg 29079  df-uhgr 29089  df-subgr 29299
This theorem is referenced by:  uhgrsubgrself  29311  isubgrsubgr  47792
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