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

Theorem cusgrafilem1 23332
Description: Lemma 1 for cusgrafi 23335. (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 23312 . . 3  |-  ( V ComplUSGrph  E  ->  ran  E  =  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
2 fveq2 5684 . . . . . . . . . 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 2975 . . . . . . . . . . 11  |-  ( N  e.  V  ->  N  e.  _V )
8 vex 2969 . . . . . . . . . . 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 12147 . . . . . . . . 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 2469 . . . . . 6  |-  ( ( ( N  e.  V  /\  x  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  x  =  {
a ,  N }
) ) )  -> 
( # `  x )  =  2 )
1514rexlimdvaa 2836 . . . . 5  |-  ( ( N  e.  V  /\  x  e.  ~P V
)  ->  ( E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } )  -> 
( # `  x )  =  2 ) )
1615ss2rabdv 3426 . . . 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 3378 . . . 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 1369    e. wcel 1756    =/= wne 2600   E.wrex 2710   {crab 2713   _Vcvv 2966    C_ wss 3321   ~Pcpw 3853   {cpr 3872   class class class wbr 4285   ran crn 4833   ` cfv 5411   2c2 10363   #chash 12095   ComplUSGrph ccusgra 23275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096  df-usgra 23211  df-cusgra 23278
This theorem is referenced by:  cusgrafi  23335
  Copyright terms: Public domain W3C validator