MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgrafilem1 Structured version   Unicode version

Theorem cusgrafilem1 23522
Description: Lemma 1 for cusgrafi 23525. (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 23502 . . 3  |-  ( V ComplUSGrph  E  ->  ran  E  =  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
2 fveq2 5789 . . . . . . . . . 10  |-  ( x  =  { a ,  N }  ->  ( # `
 x )  =  ( # `  {
a ,  N }
) )
32adantl 466 . . . . . . . . 9  |-  ( ( a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  ( # `  {
a ,  N }
) )
43adantl 466 . . . . . . . 8  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) )  ->  ( # `
 x )  =  ( # `  {
a ,  N }
) )
54adantl 466 . . . . . . 7  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  x )  =  ( # `  {
a ,  N }
) )
6 simprrl 763 . . . . . . . 8  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
a  =/=  N )
7 elex 3077 . . . . . . . . . . 11  |-  ( N  e.  V  ->  N  e.  _V )
8 vex 3071 . . . . . . . . . . 11  |-  a  e. 
_V
97, 8jctil 537 . . . . . . . . . 10  |-  ( N  e.  V  ->  (
a  e.  _V  /\  N  e.  _V )
)
109ad2antrr 725 . . . . . . . . 9  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( a  e.  _V  /\  N  e.  _V )
)
11 hashprg 12257 . . . . . . . . 9  |-  ( ( a  e.  _V  /\  N  e.  _V )  ->  ( a  =/=  N  <->  (
# `  { a ,  N } )  =  2 ) )
1210, 11syl 16 . . . . . . . 8  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( a  =/=  N  <->  (
# `  { a ,  N } )  =  2 ) )
136, 12mpbid 210 . . . . . . 7  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  { a ,  N } )  =  2 )
145, 13eqtrd 2492 . . . . . 6  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  x )  =  2 )
1514rexlimdvaa 2938 . . . . 5  |-  ( ( N  e.  V  /\  x  e.  ~P V
)  ->  ( E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  2 ) )
1615ss2rabdv 3531 . . . 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 3483 . . . 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 221 . . 3  |-  ( ran 
E  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( N  e.  V  ->  P  C_  ran  E ) )
221, 21syl 16 . 2  |-  ( V ComplUSGrph  E  ->  ( N  e.  V  ->  P  C_  ran  E ) )
2322imp 429 1  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  P  C_ 
ran  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799   _Vcvv 3068    C_ wss 3426   ~Pcpw 3958   {cpr 3977   class class class wbr 4390   ran crn 4939   ` cfv 5516   2c2 10472   #chash 12204   ComplUSGrph ccusgra 23465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-hash 12205  df-usgra 23401  df-cusgra 23468
This theorem is referenced by:  cusgrafi  23525
  Copyright terms: Public domain W3C validator