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Theorem umgran0 21343
Description: An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
umgran0  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  =/=  (/) )

Proof of Theorem umgran0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3420 . . 3  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
2 umgraf 21341 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
32ffvelrnda 5861 . . . 4  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
433impa 1148 . . 3  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
51, 4sseldi 3338 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  ( ~P V  \  { (/)
} ) )
6 eldifsni 3920 . 2  |-  ( ( E `  F )  e.  ( ~P V  \  { (/) } )  -> 
( E `  F
)  =/=  (/) )
75, 6syl 16 1  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   {crab 2701    \ cdif 3309   (/)c0 3620   ~Pcpw 3791   {csn 3806   class class class wbr 4204    Fn wfn 5440   ` cfv 5445    <_ cle 9110   2c2 10038   #chash 11606   UMGrph cumg 21335
This theorem is referenced by:  umgraex  21346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fv 5453  df-umgra 21336
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