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Theorem umgran0 23887
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 3271 . . 3  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
2 umgraf 23885 . . . . 5  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
3 ffvelrn 5679 . . . . 5  |-  ( ( E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
42, 3sylan 457 . . . 4  |-  ( ( ( V UMGrph  E  /\  E  Fn  A )  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
543impa 1146 . . 3  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
61, 5sseldi 3191 . 2  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  e.  ( ~P V  \  { (/)
} ) )
7 eldifsni 3763 . 2  |-  ( ( E `  F )  e.  ( ~P V  \  { (/) } )  -> 
( E `  F
)  =/=  (/) )
86, 7syl 15 1  |-  ( ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A
)  ->  ( E `  F )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039    Fn wfn 5266   -->wf 5267   ` cfv 5271    <_ cle 8884   2c2 9811   #chash 11353   UMGrph cumg 23875
This theorem is referenced by:  umgraex  23890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-umgra 23878
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