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Theorem uhgrass 24879
Description: An edge is a subset of vertices, analogous to umgrass 24892. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
Assertion
Ref Expression
uhgrass  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  C_  V )

Proof of Theorem uhgrass
StepHypRef Expression
1 uhgraf 24872 . . . 4  |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
21ffvelrnda 6037 . . 3  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  e.  ( ~P V  \  { (/) } ) )
32eldifad 3454 . 2  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  e.  ~P V
)
43elpwid 3995 1  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  C_  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870    \ cdif 3439    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {csn 4002   class class class wbr 4426   dom cdm 4854   ` cfv 5601   UHGrph cuhg 24863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-uhgra 24865
This theorem is referenced by: (None)
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