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

Theorem usgrasscusgra 21484
Description: An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Assertion
Ref Expression
usgrasscusgra  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  A. e  e.  ran  E E. f  e.  ran  F  e  =  f )
Distinct variable groups:    e, E    e, F, f    e, V
Allowed substitution hints:    E( f)    V( f)

Proof of Theorem usgrasscusgra
Dummy variables  a 
b  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrarnedg 21396 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
2 iscusgra0 21458 . . . . . . 7  |-  ( V ComplUSGrph  F  ->  ( V USGrph  F  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F ) )
3 simplrr 738 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  b  e.  V )
4 sneq 3817 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { k }  =  { b } )
54difeq2d 3457 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( V  \  { k } )  =  ( V 
\  { b } ) )
6 preq2 3876 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { n ,  k }  =  { n ,  b } )
76eleq1d 2501 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( { n ,  k }  e.  ran  F  <->  { n ,  b }  e.  ran  F ) )
85, 7raleqbidv 2908 . . . . . . . . . . . . 13  |-  ( k  =  b  ->  ( A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  <->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
98rspcv 3040 . . . . . . . . . . . 12  |-  ( b  e.  V  ->  ( A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  ->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
103, 9syl 16 . . . . . . . . . . 11  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  ( A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  ->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
11 simplrl 737 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  V )
12 elsn 3821 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  { b }  <-> 
a  =  b )
13 nne 2602 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  a  =/=  b  <->  a  =  b )
1412, 13bitr4i 244 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  { b }  <->  -.  a  =/=  b
)
1514biimpi 187 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  { b }  ->  -.  a  =/=  b )
1615a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  F  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( a  e.  { b }  ->  -.  a  =/=  b ) )
1716con2d 109 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  F  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( a  =/=  b  ->  -.  a  e.  { b } ) )
1817imp 419 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  -.  a  e.  { b } )
1911, 18eldifd 3323 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  ( V  \  {
b } ) )
2019adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  a  e.  ( V  \  { b } ) )
21 preq1 3875 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  { n ,  b }  =  { a ,  b } )
2221eleq1d 2501 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  ( { n ,  b }  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
2322rspcv 3040 . . . . . . . . . . . 12  |-  ( a  e.  ( V  \  { b } )  ->  ( A. n  e.  ( V  \  {
b } ) { n ,  b }  e.  ran  F  ->  { a ,  b }  e.  ran  F
) )
2420, 23syl 16 . . . . . . . . . . 11  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  ( A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F  ->  { a ,  b }  e.  ran  F
) )
25 eleq1 2495 . . . . . . . . . . . . . 14  |-  ( { a ,  b }  =  e  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
2625eqcoms 2438 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
27 equid 1688 . . . . . . . . . . . . . 14  |-  e  =  e
28 equequ2 1698 . . . . . . . . . . . . . . 15  |-  ( f  =  e  ->  (
e  =  f  <->  e  =  e ) )
2928rspcev 3044 . . . . . . . . . . . . . 14  |-  ( ( e  e.  ran  F  /\  e  =  e
)  ->  E. f  e.  ran  F  e  =  f )
3027, 29mpan2 653 . . . . . . . . . . . . 13  |-  ( e  e.  ran  F  ->  E. f  e.  ran  F  e  =  f )
3126, 30syl6bi 220 . . . . . . . . . . . 12  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  ->  E. f  e.  ran  F  e  =  f ) )
3231ad2antll 710 . . . . . . . . . . 11  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  ( {
a ,  b }  e.  ran  F  ->  E. f  e.  ran  F  e  =  f ) )
3310, 24, 323syld 53 . . . . . . . . . 10  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  ( A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  ->  E. f  e.  ran  F  e  =  f ) )
3433exp31 588 . . . . . . . . 9  |-  ( V USGrph  F  ->  ( ( a  e.  V  /\  b  e.  V )  ->  (
( a  =/=  b  /\  e  =  {
a ,  b } )  ->  ( A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  ->  E. f  e.  ran  F  e  =  f ) ) ) )
3534com24 83 . . . . . . . 8  |-  ( V USGrph  F  ->  ( A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  F  -> 
( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  (
( a  e.  V  /\  b  e.  V
)  ->  E. f  e.  ran  F  e  =  f ) ) ) )
3635imp 419 . . . . . . 7  |-  ( ( V USGrph  F  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  F )  ->  ( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  (
( a  e.  V  /\  b  e.  V
)  ->  E. f  e.  ran  F  e  =  f ) ) )
372, 36syl 16 . . . . . 6  |-  ( V ComplUSGrph  F  ->  ( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  (
( a  e.  V  /\  b  e.  V
)  ->  E. f  e.  ran  F  e  =  f ) ) )
3837com13 76 . . . . 5  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  ( V ComplUSGrph  F  ->  E. f  e.  ran  F  e  =  f ) ) )
3938rexlimivv 2827 . . . 4  |-  ( E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  e  =  { a ,  b } )  ->  ( V ComplUSGrph  F  ->  E. f  e.  ran  F  e  =  f ) )
401, 39syl 16 . . 3  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  -> 
( V ComplUSGrph  F  ->  E. f  e.  ran  F  e  =  f ) )
4140impancom 428 . 2  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( e  e. 
ran  E  ->  E. f  e.  ran  F  e  =  f ) )
4241ralrimiv 2780 1  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  A. e  e.  ran  E E. f  e.  ran  F  e  =  f )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    \ cdif 3309   {csn 3806   {cpr 3807   class class class wbr 4204   ran crn 4871   USGrph cusg 21357   ComplUSGrph ccusgra 21423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-usgra 21359  df-cusgra 21426
  Copyright terms: Public domain W3C validator