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Theorem cusgrafilem1 25149
Description: Lemma 1 for cusgrafi 25152. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
Hypothesis
Ref Expression
cusgrafi.p  |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }
Assertion
Ref Expression
cusgrafilem1  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  P  C_ 
ran  E )
Distinct variable groups:    N, a, x    V, a, x    x, E
Allowed substitution hints:    P( x, a)    E( a)

Proof of Theorem cusgrafilem1
StepHypRef Expression
1 cusgrarn 25129 . . 3  |-  ( V ComplUSGrph  E  ->  ran  E  =  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
2 fveq2 5825 . . . . . . . . . 10  |-  ( x  =  { a ,  N }  ->  ( # `
 x )  =  ( # `  {
a ,  N }
) )
32adantl 467 . . . . . . . . 9  |-  ( ( a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  ( # `  {
a ,  N }
) )
43adantl 467 . . . . . . . 8  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) )  ->  ( # `
 x )  =  ( # `  {
a ,  N }
) )
54adantl 467 . . . . . . 7  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  x )  =  ( # `  {
a ,  N }
) )
6 simprrl 772 . . . . . . . 8  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
a  =/=  N )
7 elex 3031 . . . . . . . . . . 11  |-  ( N  e.  V  ->  N  e.  _V )
8 vex 3025 . . . . . . . . . . 11  |-  a  e. 
_V
97, 8jctil 539 . . . . . . . . . 10  |-  ( N  e.  V  ->  (
a  e.  _V  /\  N  e.  _V )
)
109ad2antrr 730 . . . . . . . . 9  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( a  e.  _V  /\  N  e.  _V )
)
11 hashprg 12522 . . . . . . . . 9  |-  ( ( a  e.  _V  /\  N  e.  _V )  ->  ( a  =/=  N  <->  (
# `  { a ,  N } )  =  2 ) )
1210, 11syl 17 . . . . . . . 8  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( a  =/=  N  <->  (
# `  { a ,  N } )  =  2 ) )
136, 12mpbid 213 . . . . . . 7  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  { a ,  N } )  =  2 )
145, 13eqtrd 2462 . . . . . 6  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  x )  =  2 )
1514rexlimdvaa 2857 . . . . 5  |-  ( ( N  e.  V  /\  x  e.  ~P V
)  ->  ( E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  2 ) )
1615ss2rabdv 3485 . . . 4  |-  ( N  e.  V  ->  { x  e.  ~P V  |  E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } ) } 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } )
17 cusgrafi.p . . . . . 6  |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }
1817a1i 11 . . . . 5  |-  ( ran 
E  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  P  =  { x  e. 
~P V  |  E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } ) } )
19 id 22 . . . . 5  |-  ( ran 
E  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ran 
E  =  { x  e.  ~P V  |  (
# `  x )  =  2 } )
2018, 19sseq12d 3436 . . . 4  |-  ( ran 
E  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( P  C_  ran  E  <->  { x  e.  ~P V  |  E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } ) } 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } ) )
2116, 20syl5ibr 224 . . 3  |-  ( ran 
E  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( N  e.  V  ->  P  C_  ran  E ) )
221, 21syl 17 . 2  |-  ( V ComplUSGrph  E  ->  ( N  e.  V  ->  P  C_  ran  E ) )
2322imp 430 1  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  P  C_ 
ran  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   E.wrex 2715   {crab 2718   _Vcvv 3022    C_ wss 3379   ~Pcpw 3924   {cpr 3943   class class class wbr 4366   ran crn 4797   ` cfv 5544   2c2 10610   #chash 12465   ComplUSGrph ccusgra 25088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-hash 12466  df-usgra 25002  df-cusgra 25091
This theorem is referenced by:  cusgrafi  25152
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