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Theorem usgrasscusgra 24604
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 24505 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
2 iscusgra0 24578 . . . . . . 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 3954 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { k }  =  { b } )
54difeq2d 3536 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( V  \  { k } )  =  ( V 
\  { b } ) )
6 preq2 4024 . . . . . . . . . . . . . . 15  |-  ( k  =  b  ->  { n ,  k }  =  { n ,  b } )
76eleq1d 2451 . . . . . . . . . . . . . 14  |-  ( k  =  b  ->  ( { n ,  k }  e.  ran  F  <->  { n ,  b }  e.  ran  F ) )
85, 7raleqbidv 2993 . . . . . . . . . . . . 13  |-  ( k  =  b  ->  ( A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  F  <->  A. n  e.  ( V  \  { b } ) { n ,  b }  e.  ran  F ) )
98rspcv 3131 . . . . . . . . . . . 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 3958 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  { b }  <-> 
a  =  b )
13 nne 2583 . . . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  -.  a  e.  { b } )
1911, 18eldifd 3400 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  a  =/=  b )  ->  a  e.  ( V  \  {
b } ) )
2019adantrr 714 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  F  /\  ( a  e.  V  /\  b  e.  V
) )  /\  (
a  =/=  b  /\  e  =  { a ,  b } ) )  ->  a  e.  ( V  \  { b } ) )
21 preq1 4023 . . . . . . . . . . . . . 14  |-  ( n  =  a  ->  { n ,  b }  =  { a ,  b } )
2221eleq1d 2451 . . . . . . . . . . . . 13  |-  ( n  =  a  ->  ( { n ,  b }  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
2322rspcv 3131 . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( { a ,  b }  =  e  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
2625eqcoms 2394 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  b }  ->  ( { a ,  b }  e.  ran  F  <->  e  e.  ran  F ) )
27 equid 1799 . . . . . . . . . . . . . 14  |-  e  =  e
28 equequ2 1807 . . . . . . . . . . . . . . 15  |-  ( f  =  e  ->  (
e  =  f  <->  e  =  e ) )
2928rspcev 3135 . . . . . . . . . . . . . 14  |-  ( ( e  e.  ran  F  /\  e  =  e
)  ->  E. f  e.  ran  F  e  =  f )
3027, 29mpan2 669 . . . . . . . . . . . . 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 726 . . . . . . . . . . 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 602 . . . . . . . . 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 427 . . . . . . 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 2879 . . . 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 2794 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733    \ cdif 3386   {csn 3944   {cpr 3946   class class class wbr 4367   ran crn 4914   USGrph cusg 24451   ComplUSGrph ccusgra 24539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-hash 12308  df-usgra 24454  df-cusgra 24542
This theorem is referenced by: (None)
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