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Theorem uhgrass 24097
Description: An edge is a subset of vertices, analogous to umgrass 24110. (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 24090 . . . 4  |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
21ffvelrnda 6031 . . 3  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  e.  ( ~P V  \  { (/) } ) )
32eldifad 3493 . 2  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  e.  ~P V
)
43elpwid 4025 1  |-  ( ( V UHGrph  E  /\  F  e. 
dom  E )  -> 
( E `  F
)  C_  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    \ cdif 3478    C_ wss 3481   (/)c0 3790   ~Pcpw 4015   {csn 4032   class class class wbr 4452   dom cdm 5004   ` cfv 5593   UHGrph cuhg 24081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-uhgra 24083
This theorem is referenced by: (None)
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