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Theorem usgrasscusgra 24348
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 24249 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
2 iscusgra0 24322 . . . . . . 7  |-  ( V ComplUSGrph  F  ->  ( V USGrph  F  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F ) )
3 simplrr 760 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  b  e.  V )
4 sneq 4020 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { k }  =  { b } )
54difeq2d 3604 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( V  \  { k } )  =  ( V 
\  { b } ) )
6 preq2 4091 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { n ,  k }  =  { n ,  b } )
76eleq1d 2510 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( { n ,  k }  e.  ran  F  <->  { n ,  b }  e.  ran  F ) )
85, 7raleqbidv 3052 . . . . . . . . . . . . 13  |-  ( k  =  b  ->  ( A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  <->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
98rspcv 3190 . . . . . . . . . . . 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 759 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  V )
12 elsn 4024 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  { b }  <-> 
a  =  b )
13 nne 2642 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  a  =/=  b  <->  a  =  b )
1412, 13bitr4i 252 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  { b }  <->  -.  a  =/=  b
)
1514biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  { b }  ->  -.  a  =/=  b )
1615a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  F  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( a  e.  { b }  ->  -.  a  =/=  b ) )
1716con2d 115 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  F  /\  (
a  e.  V  /\  b  e.  V )
)  ->  ( a  =/=  b  ->  -.  a  e.  { b } ) )
1817imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  -.  a  e.  { b } )
1911, 18eldifd 3469 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  ( V  \  {
b } ) )
2019adantrr 716 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  a  e.  ( V  \  { b } ) )
21 preq1 4090 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  { n ,  b }  =  { a ,  b } )
2221eleq1d 2510 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  ( { n ,  b }  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
2322rspcv 3190 . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . 14  |-  ( { a ,  b }  =  e  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
2625eqcoms 2453 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
27 equid 1775 . . . . . . . . . . . . . 14  |-  e  =  e
28 equequ2 1783 . . . . . . . . . . . . . . 15  |-  ( f  =  e  ->  (
e  =  f  <->  e  =  e ) )
2928rspcev 3194 . . . . . . . . . . . . . 14  |-  ( ( e  e.  ran  F  /\  e  =  e
)  ->  E. f  e.  ran  F  e  =  f )
3027, 29mpan2 671 . . . . . . . . . . . . 13  |-  ( e  e.  ran  F  ->  E. f  e.  ran  F  e  =  f )
3126, 30syl6bi 228 . . . . . . . . . . . 12  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  ->  E. f  e.  ran  F  e  =  f ) )
3231ad2antll 728 . . . . . . . . . . 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 55 . . . . . . . . . 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 604 . . . . . . . . 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 87 . . . . . . . 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 429 . . . . . . 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 80 . . . . 5  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  ( V ComplUSGrph  F  ->  E. f  e.  ran  F  e  =  f ) ) )
3938rexlimivv 2938 . . . 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 440 . 2  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( e  e. 
ran  E  ->  E. f  e.  ran  F  e  =  f ) )
4241ralrimiv 2853 1  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  A. e  e.  ran  E E. f  e.  ran  F  e  =  f )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   E.wrex 2792    \ cdif 3455   {csn 4010   {cpr 4012   class class class wbr 4433   ran crn 4986   USGrph cusg 24195   ComplUSGrph ccusgra 24283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  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 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-hash 12380  df-usgra 24198  df-cusgra 24286
This theorem is referenced by: (None)
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