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Theorem umgran0 21386
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 3414 . . 3  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
2 umgraf 21384 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
32ffvelrnda 5899 . . . 4  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
433impa 1149 . . 3  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
51, 4sseldi 3332 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  ( ~P V  \  { (/)
} ) )
6 eldifsni 3952 . 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 360    /\ w3a 937    e. wcel 1727    =/= wne 2605   {crab 2715    \ cdif 3303   (/)c0 3613   ~Pcpw 3823   {csn 3838   class class class wbr 4237    Fn wfn 5478   ` cfv 5483    <_ cle 9152   2c2 10080   #chash 11649   UMGrph cumg 21378
This theorem is referenced by:  umgraex  21389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-umgra 21379
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