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Theorem sizeusglecusglem1 23543
Description: Lemma 1 for sizeusglecusg 23545. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
Assertion
Ref Expression
sizeusglecusglem1  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  (  _I  |`  ran  E
) : ran  E -1-1-> ran 
F )

Proof of Theorem sizeusglecusglem1
Dummy variables  a 
b  e  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5783 . . 3  |-  (  _I  |`  ran  E ) : ran  E -1-1-onto-> ran  E
2 f1of1 5747 . . 3  |-  ( (  _I  |`  ran  E ) : ran  E -1-1-onto-> ran  E  ->  (  _I  |`  ran  E
) : ran  E -1-1-> ran 
E )
31, 2ax-mp 5 . 2  |-  (  _I  |`  ran  E ) : ran  E -1-1-> ran  E
4 usgrarnedg 23454 . . . . 5  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
5 iscusgra0 23516 . . . . . 6  |-  ( V ComplUSGrph  F  ->  ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  { x }
) { y ,  x }  e.  ran  F ) )
6 simprr 756 . . . . . . . . . . . . . . . 16  |-  ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  ->  b  e.  V )
7 eldifsn 4107 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  e.  ( V  \  { b } )  <-> 
( a  e.  V  /\  a  =/=  b
) )
87simplbi2 625 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  V  ->  (
a  =/=  b  -> 
a  e.  ( V 
\  { b } ) ) )
98adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( a  =/=  b  ->  a  e.  ( V 
\  { b } ) ) )
109impcom 430 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  ->  a  e.  ( V  \  {
b } ) )
1110adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
a  e.  ( V 
\  { b } ) )
12 sneq 3994 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  b  ->  { x }  =  { b } )
1312difeq2d 3581 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  ( V  \  { x }
)  =  ( V 
\  { b } ) )
1413eleq2d 2524 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  (
a  e.  ( V 
\  { x }
)  <->  a  e.  ( V  \  { b } ) ) )
1514adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
( a  e.  ( V  \  { x } )  <->  a  e.  ( V  \  { b } ) ) )
1611, 15mpbird 232 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
a  e.  ( V 
\  { x }
) )
17 preq12 4063 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  =  a  /\  x  =  b )  ->  { y ,  x }  =  { a ,  b } )
1817expcom 435 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
y  =  a  ->  { y ,  x }  =  { a ,  b } ) )
1918adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
( y  =  a  ->  { y ,  x }  =  {
a ,  b } ) )
2019imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V ) )  /\  x  =  b )  /\  y  =  a
)  ->  { y ,  x }  =  {
a ,  b } )
2120eleq1d 2523 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V ) )  /\  x  =  b )  /\  y  =  a
)  ->  ( {
y ,  x }  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
2216, 21rspcdv 3180 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
( A. y  e.  ( V  \  {
x } ) { y ,  x }  e.  ran  F  ->  { a ,  b }  e.  ran  F ) )
236, 22rspcimdv 3178 . . . . . . . . . . . . . . 15  |-  ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  ->  ( A. x  e.  V  A. y  e.  ( V  \  { x }
) { y ,  x }  e.  ran  F  ->  { a ,  b }  e.  ran  F ) )
2423ex 434 . . . . . . . . . . . . . 14  |-  ( a  =/=  b  ->  (
( a  e.  V  /\  b  e.  V
)  ->  ( A. x  e.  V  A. y  e.  ( V  \  { x } ) { y ,  x }  e.  ran  F  ->  { a ,  b }  e.  ran  F
) ) )
2524com13 80 . . . . . . . . . . . . 13  |-  ( A. x  e.  V  A. y  e.  ( V  \  { x } ) { y ,  x }  e.  ran  F  -> 
( ( a  e.  V  /\  b  e.  V )  ->  (
a  =/=  b  ->  { a ,  b }  e.  ran  F
) ) )
2625adantl 466 . . . . . . . . . . . 12  |-  ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  {
x } ) { y ,  x }  e.  ran  F )  -> 
( ( a  e.  V  /\  b  e.  V )  ->  (
a  =/=  b  ->  { a ,  b }  e.  ran  F
) ) )
2726imp 429 . . . . . . . . . . 11  |-  ( ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  { x } ) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V ) )  -> 
( a  =/=  b  ->  { a ,  b }  e.  ran  F
) )
2827com12 31 . . . . . . . . . 10  |-  ( a  =/=  b  ->  (
( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  { x }
) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V ) )  ->  { a ,  b }  e.  ran  F
) )
29 eleq1 2526 . . . . . . . . . . 11  |-  ( e  =  { a ,  b }  ->  (
e  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
3029imbi2d 316 . . . . . . . . . 10  |-  ( e  =  { a ,  b }  ->  (
( ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  {
x } ) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V
) )  ->  e  e.  ran  F )  <->  ( (
( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  { x } ) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V ) )  ->  { a ,  b }  e.  ran  F
) ) )
3128, 30syl5ibr 221 . . . . . . . . 9  |-  ( e  =  { a ,  b }  ->  (
a  =/=  b  -> 
( ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  {
x } ) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V
) )  ->  e  e.  ran  F ) ) )
3231impcom 430 . . . . . . . 8  |-  ( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  ( ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  {
x } ) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V
) )  ->  e  e.  ran  F ) )
3332com12 31 . . . . . . 7  |-  ( ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  { x } ) { y ,  x }  e.  ran  F )  /\  ( a  e.  V  /\  b  e.  V ) )  -> 
( ( a  =/=  b  /\  e  =  { a ,  b } )  ->  e  e.  ran  F ) )
3433rexlimdvva 2952 . . . . . 6  |-  ( ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  {
x } ) { y ,  x }  e.  ran  F )  -> 
( E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  { a ,  b } )  ->  e  e.  ran  F ) )
355, 34syl 16 . . . . 5  |-  ( V ComplUSGrph  F  ->  ( E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  { a ,  b } )  ->  e  e.  ran  F ) )
364, 35syl5com 30 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  -> 
( V ComplUSGrph  F  ->  e  e.  ran  F ) )
3736impancom 440 . . 3  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( e  e. 
ran  E  ->  e  e. 
ran  F ) )
3837ssrdv 3469 . 2  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ran  E  C_  ran  F )
39 f1ss 5718 . 2  |-  ( ( (  _I  |`  ran  E
) : ran  E -1-1-> ran 
E  /\  ran  E  C_  ran  F )  ->  (  _I  |`  ran  E ) : ran  E -1-1-> ran  F )
403, 38, 39sylancr 663 1  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  (  _I  |`  ran  E
) : ran  E -1-1-> ran 
F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   E.wrex 2799    \ cdif 3432    C_ wss 3435   {csn 3984   {cpr 3986   class class class wbr 4399    _I cid 4738   ran crn 4948    |` cres 4949   -1-1->wf1 5522   -1-1-onto->wf1o 5524   USGrph cusg 23415   ComplUSGrph ccusgra 23481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-hash 12220  df-usgra 23417  df-cusgra 23484
This theorem is referenced by:  sizeusglecusg  23545
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