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Theorem uhgrissubgr 39493
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  |-  ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  ( S SubGraph  G  <->  ( V  C_  A  /\  I  C_  B ) ) )

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 2462 . . . 4  |-  (Edg `  S )  =  (Edg
`  S )
61, 2, 3, 4, 5subgrprop2 39492 . . 3  |-  ( S SubGraph  G  ->  ( V  C_  A  /\  I  C_  B  /\  (Edg `  S )  C_ 
~P V ) )
7 3simpa 1011 . . 3  |-  ( ( V  C_  A  /\  I  C_  B  /\  (Edg `  S )  C_  ~P V )  ->  ( V  C_  A  /\  I  C_  B ) )
86, 7syl 17 . 2  |-  ( S SubGraph  G  ->  ( V  C_  A  /\  I  C_  B
) )
9 simprl 769 . . . 4  |-  ( ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  /\  ( V 
C_  A  /\  I  C_  B ) )  ->  V  C_  A )
10 simp2 1015 . . . . . 6  |-  ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  Fun  B )
11 simpr 467 . . . . . 6  |-  ( ( V  C_  A  /\  I  C_  B )  ->  I  C_  B )
12 funssres 5645 . . . . . 6  |-  ( ( Fun  B  /\  I  C_  B )  ->  ( B  |`  dom  I )  =  I )
1310, 11, 12syl2an 484 . . . . 5  |-  ( ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  /\  ( V 
C_  A  /\  I  C_  B ) )  -> 
( B  |`  dom  I
)  =  I )
1413eqcomd 2468 . . . 4  |-  ( ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  /\  ( V 
C_  A  /\  I  C_  B ) )  ->  I  =  ( B  |` 
dom  I ) )
15 edguhgr 39365 . . . . . . . . 9  |-  ( ( S  e. UHGraph  /\  e  e.  (Edg `  S )
)  ->  e  e.  ~P (Vtx `  S )
)
1615ex 440 . . . . . . . 8  |-  ( S  e. UHGraph  ->  ( e  e.  (Edg `  S )  ->  e  e.  ~P (Vtx `  S ) ) )
171pweqi 3967 . . . . . . . . 9  |-  ~P V  =  ~P (Vtx `  S
)
1817eleq2i 2532 . . . . . . . 8  |-  ( e  e.  ~P V  <->  e  e.  ~P (Vtx `  S )
)
1916, 18syl6ibr 235 . . . . . . 7  |-  ( S  e. UHGraph  ->  ( e  e.  (Edg `  S )  ->  e  e.  ~P V
) )
2019ssrdv 3450 . . . . . 6  |-  ( S  e. UHGraph  ->  (Edg `  S
)  C_  ~P V
)
21203ad2ant3 1037 . . . . 5  |-  ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  (Edg `  S
)  C_  ~P V
)
2221adantr 471 . . . 4  |-  ( ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  /\  ( V 
C_  A  /\  I  C_  B ) )  -> 
(Edg `  S )  C_ 
~P V )
231, 2, 3, 4, 5issubgr 39489 . . . . . 6  |-  ( ( G  e.  W  /\  S  e. UHGraph  )  ->  ( S SubGraph  G  <->  ( V  C_  A  /\  I  =  ( B  |`  dom  I )  /\  (Edg `  S
)  C_  ~P V
) ) )
24233adant2 1033 . . . . 5  |-  ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  ( S SubGraph  G  <->  ( V  C_  A  /\  I  =  ( B  |`  dom  I
)  /\  (Edg `  S
)  C_  ~P V
) ) )
2524adantr 471 . . . 4  |-  ( ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  /\  ( V 
C_  A  /\  I  C_  B ) )  -> 
( S SubGraph  G  <->  ( V  C_  A  /\  I  =  ( B  |`  dom  I
)  /\  (Edg `  S
)  C_  ~P V
) ) )
269, 14, 22, 25mpbir3and 1197 . . 3  |-  ( ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  /\  ( V 
C_  A  /\  I  C_  B ) )  ->  S SubGraph  G )
2726ex 440 . 2  |-  ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  ( ( V 
C_  A  /\  I  C_  B )  ->  S SubGraph  G ) )
288, 27impbid2 209 1  |-  ( ( G  e.  W  /\  Fun  B  /\  S  e. UHGraph  )  ->  ( S SubGraph  G  <->  ( V  C_  A  /\  I  C_  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    C_ wss 3416   ~Pcpw 3963   class class class wbr 4418   dom cdm 4856    |` cres 4858   Fun wfun 5599   ` cfv 5605  Vtxcvtx 39243  iEdgciedg 39244   UHGraph cuhgr 39289  Edgcedga 39354   SubGraph csubgr 39485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fv 5613  df-uhgr 39291  df-edga 39355  df-subgr 39486
This theorem is referenced by:  uhgrsubgrself  39498
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