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Theorem usgrasscusgra 21445
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 21357 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
2 iscusgra0 21419 . . . . . . 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 3785 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { k }  =  { b } )
54difeq2d 3425 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( V  \  { k } )  =  ( V 
\  { b } ) )
6 preq2 3844 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { n ,  k }  =  { n ,  b } )
76eleq1d 2470 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( { n ,  k }  e.  ran  F  <->  { n ,  b }  e.  ran  F ) )
85, 7raleqbidv 2876 . . . . . . . . . . . . 13  |-  ( k  =  b  ->  ( A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  <->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
98rspcv 3008 . . . . . . . . . . . 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 3789 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  { b }  <-> 
a  =  b )
13 nne 2571 . . . . . . . . . . . . . . . . . . 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 3291 . . . . . . . . . . . . 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 3843 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  { n ,  b }  =  { a ,  b } )
2221eleq1d 2470 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  ( { n ,  b }  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
2322rspcv 3008 . . . . . . . . . . . 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 2464 . . . . . . . . . . . . . 14  |-  ( { a ,  b }  =  e  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
2625eqcoms 2407 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
27 equid 1684 . . . . . . . . . . . . . 14  |-  e  =  e
28 equequ2 1694 . . . . . . . . . . . . . . 15  |-  ( f  =  e  ->  (
e  =  f  <->  e  =  e ) )
2928rspcev 3012 . . . . . . . . . . . . . 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 2795 . . . 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 2748 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277   {csn 3774   {cpr 3775   class class class wbr 4172   ran crn 4838   USGrph cusg 21318   ComplUSGrph ccusgra 21384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-usgra 21320  df-cusgra 21387
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