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Theorem upgredg 39389
Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)
Hypotheses
Ref Expression
upgredg.v  |-  V  =  (Vtx `  G )
upgredg.e  |-  E  =  (Edg `  G )
Assertion
Ref Expression
upgredg  |-  ( ( G  e. UPGraph  /\  C  e.  E )  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } )
Distinct variable groups:    C, a,
b    G, a, b    V, a, b
Allowed substitution hints:    E( a, b)

Proof of Theorem upgredg
Dummy variables  x  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 upgredg.e . . . . . . 7  |-  E  =  (Edg `  G )
2 edgaval 39373 . . . . . . 7  |-  ( G  e. UPGraph  ->  (Edg `  G
)  =  ran  (iEdg `  G ) )
31, 2syl5eq 2517 . . . . . 6  |-  ( G  e. UPGraph  ->  E  =  ran  (iEdg `  G ) )
43eleq2d 2534 . . . . 5  |-  ( G  e. UPGraph  ->  ( C  e.  E  <->  C  e.  ran  (iEdg `  G ) ) )
5 upgredg.v . . . . . . . 8  |-  V  =  (Vtx `  G )
6 eqid 2471 . . . . . . . 8  |-  (iEdg `  G )  =  (iEdg `  G )
75, 6upgrf 39332 . . . . . . 7  |-  ( G  e. UPGraph  ->  (iEdg `  G
) : dom  (iEdg `  G ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
8 frn 5747 . . . . . . 7  |-  ( (iEdg `  G ) : dom  (iEdg `  G ) --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }  ->  ran  (iEdg `  G
)  C_  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
97, 8syl 17 . . . . . 6  |-  ( G  e. UPGraph  ->  ran  (iEdg `  G
)  C_  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
109sseld 3417 . . . . 5  |-  ( G  e. UPGraph  ->  ( C  e. 
ran  (iEdg `  G )  ->  C  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
114, 10sylbid 223 . . . 4  |-  ( G  e. UPGraph  ->  ( C  e.  E  ->  C  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } ) )
1211imp 436 . . 3  |-  ( ( G  e. UPGraph  /\  C  e.  E )  ->  C  e.  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
13 fveq2 5879 . . . . . 6  |-  ( x  =  C  ->  ( # `
 x )  =  ( # `  C
) )
1413breq1d 4405 . . . . 5  |-  ( x  =  C  ->  (
( # `  x )  <_  2  <->  ( # `  C
)  <_  2 ) )
1514elrab 3184 . . . 4  |-  ( C  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  <_  2
) )
16 2nn0 10910 . . . . . . 7  |-  2  e.  NN0
17 hashbnd 12559 . . . . . . 7  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  2  e. 
NN0  /\  ( # `  C
)  <_  2 )  ->  C  e.  Fin )
1816, 17mp3an2 1378 . . . . . 6  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  <_  2
)  ->  C  e.  Fin )
19 hashcl 12576 . . . . . 6  |-  ( C  e.  Fin  ->  ( # `
 C )  e. 
NN0 )
2018, 19syl 17 . . . . 5  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  <_  2
)  ->  ( # `  C
)  e.  NN0 )
21 nn0re 10902 . . . . . . . . 9  |-  ( (
# `  C )  e.  NN0  ->  ( # `  C
)  e.  RR )
22 2re 10701 . . . . . . . . . 10  |-  2  e.  RR
2322a1i 11 . . . . . . . . 9  |-  ( (
# `  C )  e.  NN0  ->  2  e.  RR )
2421, 23leloed 9795 . . . . . . . 8  |-  ( (
# `  C )  e.  NN0  ->  ( ( # `
 C )  <_ 
2  <->  ( ( # `  C )  <  2  \/  ( # `  C
)  =  2 ) ) )
2524adantl 473 . . . . . . 7  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  <_  2  <->  ( ( # `  C
)  <  2  \/  ( # `  C )  =  2 ) ) )
26 nn0lt2 11023 . . . . . . . . . . 11  |-  ( ( ( # `  C
)  e.  NN0  /\  ( # `  C )  <  2 )  -> 
( ( # `  C
)  =  0  \/  ( # `  C
)  =  1 ) )
2726ex 441 . . . . . . . . . 10  |-  ( (
# `  C )  e.  NN0  ->  ( ( # `
 C )  <  2  ->  ( ( # `
 C )  =  0  \/  ( # `  C )  =  1 ) ) )
2827adantl 473 . . . . . . . . 9  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  <  2  ->  ( ( # `  C
)  =  0  \/  ( # `  C
)  =  1 ) ) )
29 eldifsn 4088 . . . . . . . . . . . 12  |-  ( C  e.  ( ~P V  \  { (/) } )  <->  ( C  e.  ~P V  /\  C  =/=  (/) ) )
30 hasheq0 12582 . . . . . . . . . . . . . 14  |-  ( C  e.  ~P V  -> 
( ( # `  C
)  =  0  <->  C  =  (/) ) )
3130adantr 472 . . . . . . . . . . . . 13  |-  ( ( C  e.  ~P V  /\  C  =/=  (/) )  -> 
( ( # `  C
)  =  0  <->  C  =  (/) ) )
32 eqneqall 2654 . . . . . . . . . . . . . . 15  |-  ( C  =  (/)  ->  ( C  =/=  (/)  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
3332com12 31 . . . . . . . . . . . . . 14  |-  ( C  =/=  (/)  ->  ( C  =  (/)  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
3433adantl 473 . . . . . . . . . . . . 13  |-  ( ( C  e.  ~P V  /\  C  =/=  (/) )  -> 
( C  =  (/)  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
3531, 34sylbid 223 . . . . . . . . . . . 12  |-  ( ( C  e.  ~P V  /\  C  =/=  (/) )  -> 
( ( # `  C
)  =  0  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V
)  /\  C  =  { a ,  b } ) ) )
3629, 35sylbi 200 . . . . . . . . . . 11  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( ( # `  C
)  =  0  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V
)  /\  C  =  { a ,  b } ) ) )
3736adantr 472 . . . . . . . . . 10  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  =  0  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
38 hash1snb 12634 . . . . . . . . . . . 12  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( ( # `  C
)  =  1  <->  E. c  C  =  {
c } ) )
39 eleq1 2537 . . . . . . . . . . . . . . . . . . . . 21  |-  ( C  =  { c }  ->  ( C  e. 
~P V  <->  { c }  e.  ~P V
) )
40 vex 3034 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  c  e. 
_V
4140snelpw 4646 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  e.  V  <->  { c }  e.  ~P V
)
4241biimpri 211 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( { c }  e.  ~P V  ->  c  e.  V
)
4342a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( C  =  { c }  ->  ( { c }  e.  ~P V  ->  c  e.  V ) )
4439, 43sylbid 223 . . . . . . . . . . . . . . . . . . . 20  |-  ( C  =  { c }  ->  ( C  e. 
~P V  ->  c  e.  V ) )
4544com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( C  e.  ~P V  -> 
( C  =  {
c }  ->  c  e.  V ) )
4645adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( C  e.  ~P V  /\  C  =/=  (/) )  -> 
( C  =  {
c }  ->  c  e.  V ) )
4729, 46sylbi 200 . . . . . . . . . . . . . . . . 17  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( C  =  {
c }  ->  c  e.  V ) )
4847imp 436 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  C  =  { c } )  ->  c  e.  V
)
49 id 22 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  { c }  ->  C  =  {
c } )
50 dfsn2 3972 . . . . . . . . . . . . . . . . . 18  |-  { c }  =  { c ,  c }
5149, 50syl6eq 2521 . . . . . . . . . . . . . . . . 17  |-  ( C  =  { c }  ->  C  =  {
c ,  c } )
5251adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  C  =  { c } )  ->  C  =  {
c ,  c } )
53 preq1 4042 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  { a ,  b }  =  { c ,  b } )
5453eqeq2d 2481 . . . . . . . . . . . . . . . . 17  |-  ( a  =  c  ->  ( C  =  { a ,  b }  <->  C  =  { c ,  b } ) )
55 preq2 4043 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  c  ->  { c ,  b }  =  { c ,  c } )
5655eqeq2d 2481 . . . . . . . . . . . . . . . . 17  |-  ( b  =  c  ->  ( C  =  { c ,  b }  <->  C  =  { c ,  c } ) )
5754, 56rspc2ev 3149 . . . . . . . . . . . . . . . 16  |-  ( ( c  e.  V  /\  c  e.  V  /\  C  =  { c ,  c } )  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } )
5848, 48, 52, 57syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  C  =  { c } )  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } )
59 r2ex 2901 . . . . . . . . . . . . . . 15  |-  ( E. a  e.  V  E. b  e.  V  C  =  { a ,  b }  <->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) )
6058, 59sylib 201 . . . . . . . . . . . . . 14  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  C  =  { c } )  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) )
6160ex 441 . . . . . . . . . . . . 13  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( C  =  {
c }  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
6261exlimdv 1787 . . . . . . . . . . . 12  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( E. c  C  =  { c }  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
6338, 62sylbid 223 . . . . . . . . . . 11  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( ( # `  C
)  =  1  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V
)  /\  C  =  { a ,  b } ) ) )
6463adantr 472 . . . . . . . . . 10  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  =  1  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
6537, 64jaod 387 . . . . . . . . 9  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( (
# `  C )  =  0  \/  ( # `
 C )  =  1 )  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
6628, 65syld 44 . . . . . . . 8  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  <  2  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
67 hash2sspr 12685 . . . . . . . . . . . . 13  |-  ( ( C  e.  ~P V  /\  ( # `  C
)  =  2 )  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } )
6867ex 441 . . . . . . . . . . . 12  |-  ( C  e.  ~P V  -> 
( ( # `  C
)  =  2  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } ) )
6968adantr 472 . . . . . . . . . . 11  |-  ( ( C  e.  ~P V  /\  C  =/=  (/) )  -> 
( ( # `  C
)  =  2  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } ) )
7029, 69sylbi 200 . . . . . . . . . 10  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( ( # `  C
)  =  2  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } ) )
7170, 59syl6ib 234 . . . . . . . . 9  |-  ( C  e.  ( ~P V  \  { (/) } )  -> 
( ( # `  C
)  =  2  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V
)  /\  C  =  { a ,  b } ) ) )
7271adantr 472 . . . . . . . 8  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  =  2  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
7366, 72jaod 387 . . . . . . 7  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( (
# `  C )  <  2  \/  ( # `  C )  =  2 )  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
7425, 73sylbid 223 . . . . . 6  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  e.  NN0 )  ->  ( ( # `  C )  <_  2  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
7574impancom 447 . . . . 5  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  <_  2
)  ->  ( ( # `
 C )  e. 
NN0  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) ) )
7620, 75mpd 15 . . . 4  |-  ( ( C  e.  ( ~P V  \  { (/) } )  /\  ( # `  C )  <_  2
)  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) )
7715, 76sylbi 200 . . 3  |-  ( C  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  E. a E. b
( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) )
7812, 77syl 17 . 2  |-  ( ( G  e. UPGraph  /\  C  e.  E )  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  C  =  { a ,  b } ) )
7978, 59sylibr 217 1  |-  ( ( G  e. UPGraph  /\  C  e.  E )  ->  E. a  e.  V  E. b  e.  V  C  =  { a ,  b } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760    \ cdif 3387    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   {cpr 3961   class class class wbr 4395   dom cdm 4839   ran crn 4840   -->wf 5585   ` cfv 5589   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558    < clt 9693    <_ cle 9694   2c2 10681   NN0cn0 10893   #chash 12553  Vtxcvtx 39251  iEdgciedg 39252   UPGraph cupgr 39326  Edgcedga 39371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-upgr 39328  df-edga 39372
This theorem is referenced by:  upgredg2vtx  39392  upgredgpr  39393
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