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Theorem usgrasscusgra 23310
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 23222 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
2 iscusgra0 23284 . . . . . . 7  |-  ( V ComplUSGrph  F  ->  ( V USGrph  F  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F ) )
3 simplrr 755 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  b  e.  V )
4 sneq 3884 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { k }  =  { b } )
54difeq2d 3471 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( V  \  { k } )  =  ( V 
\  { b } ) )
6 preq2 3952 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { n ,  k }  =  { n ,  b } )
76eleq1d 2507 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( { n ,  k }  e.  ran  F  <->  { n ,  b }  e.  ran  F ) )
85, 7raleqbidv 2929 . . . . . . . . . . . . 13  |-  ( k  =  b  ->  ( A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  <->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
98rspcv 3066 . . . . . . . . . . . 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 754 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  V )
12 elsn 3888 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  { b }  <-> 
a  =  b )
13 nne 2610 . . . . . . . . . . . . . . . . . . 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 3336 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  ( V  \  {
b } ) )
2019adantrr 711 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  a  e.  ( V  \  { b } ) )
21 preq1 3951 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  { n ,  b }  =  { a ,  b } )
2221eleq1d 2507 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  ( { n ,  b }  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
2322rspcv 3066 . . . . . . . . . . . 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 2501 . . . . . . . . . . . . . 14  |-  ( { a ,  b }  =  e  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
2625eqcoms 2444 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
27 equid 1734 . . . . . . . . . . . . . 14  |-  e  =  e
28 equequ2 1742 . . . . . . . . . . . . . . 15  |-  ( f  =  e  ->  (
e  =  f  <->  e  =  e ) )
2928rspcev 3070 . . . . . . . . . . . . . 14  |-  ( ( e  e.  ran  F  /\  e  =  e
)  ->  E. f  e.  ran  F  e  =  f )
3027, 29mpan2 666 . . . . . . . . . . . . 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 723 . . . . . . . . . . 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 601 . . . . . . . . 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 2844 . . . 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 438 . 2  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( e  e. 
ran  E  ->  E. f  e.  ran  F  e  =  f ) )
4241ralrimiv 2796 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714    \ cdif 3322   {csn 3874   {cpr 3876   class class class wbr 4289   ran crn 4837   USGrph cusg 23183   ComplUSGrph ccusgra 23249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100  df-usgra 23185  df-cusgra 23252
This theorem is referenced by: (None)
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