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Theorem cusgrafilem1 24600
Description: Lemma 1 for cusgrafi 24603. (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 24580 . . 3  |-  ( V ComplUSGrph  E  ->  ran  E  =  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
2 fveq2 5774 . . . . . . . . . 10  |-  ( x  =  { a ,  N }  ->  ( # `
 x )  =  ( # `  {
a ,  N }
) )
32adantl 464 . . . . . . . . 9  |-  ( ( a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  ( # `  {
a ,  N }
) )
43adantl 464 . . . . . . . 8  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) )  ->  ( # `
 x )  =  ( # `  {
a ,  N }
) )
54adantl 464 . . . . . . 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 3043 . . . . . . . . . . 11  |-  ( N  e.  V  ->  N  e.  _V )
8 vex 3037 . . . . . . . . . . 11  |-  a  e. 
_V
97, 8jctil 535 . . . . . . . . . 10  |-  ( N  e.  V  ->  (
a  e.  _V  /\  N  e.  _V )
)
109ad2antrr 723 . . . . . . . . 9  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( a  e.  _V  /\  N  e.  _V )
)
11 hashprg 12364 . . . . . . . . 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 2423 . . . . . 6  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  x )  =  2 )
1514rexlimdvaa 2875 . . . . 5  |-  ( ( N  e.  V  /\  x  e.  ~P V
)  ->  ( E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  2 ) )
1615ss2rabdv 3495 . . . 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 3446 . . . 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 427 1  |-  ( ( V ComplUSGrph  E  /\  N  e.  V )  ->  P  C_ 
ran  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733   {crab 2736   _Vcvv 3034    C_ wss 3389   ~Pcpw 3927   {cpr 3946   class class class wbr 4367   ran crn 4914   ` cfv 5496   2c2 10502   #chash 12307   ComplUSGrph ccusgra 24539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-hash 12308  df-usgra 24454  df-cusgra 24542
This theorem is referenced by:  cusgrafi  24603
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