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Theorem sizeusglecusglem1 24888
Description: Lemma 1 for sizeusglecusg 24890. (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 5833 . . 3  |-  (  _I  |`  ran  E ) : ran  E -1-1-onto-> ran  E
2 f1of1 5797 . . 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 24788 . . . . 5  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  {
a ,  b } ) )
5 iscusgra0 24861 . . . . . 6  |-  ( V ComplUSGrph  F  ->  ( V USGrph  F  /\  A. x  e.  V  A. y  e.  ( V  \  { x }
) { y ,  x }  e.  ran  F ) )
6 simprr 758 . . . . . . . . . . . . . . . 16  |-  ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  ->  b  e.  V )
7 eldifsn 4096 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  e.  ( V  \  { b } )  <-> 
( a  e.  V  /\  a  =/=  b
) )
87simplbi2 623 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  V  ->  (
a  =/=  b  -> 
a  e.  ( V 
\  { b } ) ) )
98adantr 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( a  =/=  b  ->  a  e.  ( V 
\  { b } ) ) )
109impcom 428 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  ->  a  e.  ( V  \  {
b } ) )
1110adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
a  e.  ( V 
\  { b } ) )
12 sneq 3981 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  b  ->  { x }  =  { b } )
1312difeq2d 3560 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  ( V  \  { x }
)  =  ( V 
\  { b } ) )
1413eleq2d 2472 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  (
a  e.  ( V 
\  { x }
)  <->  a  e.  ( V  \  { b } ) ) )
1514adantl 464 . . . . . . . . . . . . . . . . . 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 4052 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  =  a  /\  x  =  b )  ->  { y ,  x }  =  { a ,  b } )
1817expcom 433 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
y  =  a  ->  { y ,  x }  =  { a ,  b } ) )
1918adantl 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V
) )  /\  x  =  b )  -> 
( y  =  a  ->  { y ,  x }  =  {
a ,  b } ) )
2019imp 427 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  =/=  b  /\  ( a  e.  V  /\  b  e.  V ) )  /\  x  =  b )  /\  y  =  a
)  ->  { y ,  x }  =  {
a ,  b } )
2120eleq1d 2471 . . . . . . . . . . . . . . . . 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 3162 . . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . 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 464 . . . . . . . . . . . 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 427 . . . . . . . . . . 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 29 . . . . . . . . . 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 2474 . . . . . . . . . . 11  |-  ( e  =  { a ,  b }  ->  (
e  e.  ran  F  <->  { a ,  b }  e.  ran  F ) )
3029imbi2d 314 . . . . . . . . . 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 428 . . . . . . . 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 29 . . . . . . 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 2902 . . . . . 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 17 . . . . 5  |-  ( V ComplUSGrph  F  ->  ( E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  e  =  { a ,  b } )  ->  e  e.  ran  F ) )
364, 35syl5com 28 . . . 4  |-  ( ( V USGrph  E  /\  e  e.  ran  E )  -> 
( V ComplUSGrph  F  ->  e  e.  ran  F ) )
3736impancom 438 . . 3  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( e  e. 
ran  E  ->  e  e. 
ran  F ) )
3837ssrdv 3447 . 2  |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ran  E  C_  ran  F )
39 f1ss 5768 . 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 661 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754    \ cdif 3410    C_ wss 3413   {csn 3971   {cpr 3973   class class class wbr 4394    _I cid 4732   ran crn 4823    |` cres 4824   -1-1->wf1 5565   -1-1-onto->wf1o 5567   USGrph cusg 24734   ComplUSGrph ccusgra 24822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-hash 12451  df-usgra 24737  df-cusgra 24825
This theorem is referenced by:  sizeusglecusg  24890
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