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Theorem uhgrissubgr 27043
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 2821 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 27042 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
7 3simpa 1145 . . 3 ((𝑉𝐴𝐼𝐵 ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉) → (𝑉𝐴𝐼𝐵))
86, 7syl 17 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵))
9 simprl 770 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝑉𝐴)
10 simp2 1134 . . . . . 6 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → Fun 𝐵)
11 simpr 488 . . . . . 6 ((𝑉𝐴𝐼𝐵) → 𝐼𝐵)
12 funssres 6371 . . . . . 6 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1310, 11, 12syl2an 598 . . . . 5 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (𝐵 ↾ dom 𝐼) = 𝐼)
1413eqcomd 2827 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝐼 = (𝐵 ↾ dom 𝐼))
15 edguhgr 26900 . . . . . . . . 9 ((𝑆 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝑆)) → 𝑒 ∈ 𝒫 (Vtx‘𝑆))
1615ex 416 . . . . . . . 8 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
171pweqi 4530 . . . . . . . . 9 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
1817eleq2i 2903 . . . . . . . 8 (𝑒 ∈ 𝒫 𝑉𝑒 ∈ 𝒫 (Vtx‘𝑆))
1916, 18syl6ibr 255 . . . . . . 7 (𝑆 ∈ UHGraph → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 𝑉))
2019ssrdv 3949 . . . . . 6 (𝑆 ∈ UHGraph → (Edg‘𝑆) ⊆ 𝒫 𝑉)
21203ad2ant3 1132 . . . . 5 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
2221adantr 484 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
231, 2, 3, 4, 5issubgr 27039 . . . . . 6 ((𝐺𝑊𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
24233adant2 1128 . . . . 5 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
2524adantr 484 . . . 4 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)))
269, 14, 22, 25mpbir3and 1339 . . 3 (((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) ∧ (𝑉𝐴𝐼𝐵)) → 𝑆 SubGraph 𝐺)
2726ex 416 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → ((𝑉𝐴𝐼𝐵) → 𝑆 SubGraph 𝐺))
288, 27impbid2 229 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wss 3910  𝒫 cpw 4512   class class class wbr 5039  dom cdm 5528  cres 5530  Fun wfun 6322  cfv 6328  Vtxcvtx 26767  iEdgciedg 26768  Edgcedg 26818  UHGraphcuhgr 26827   SubGraph csubgr 27035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-edg 26819  df-uhgr 26829  df-subgr 27036
This theorem is referenced by:  uhgrsubgrself  27048
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