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Theorem frgra3vlem2 25741
Description: Lemma 2 for frgra3v 25742. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra3vlem2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  ( { A ,  B ,  C } USGrph  E  ->  ( E! x  e.  { A ,  B ,  C }  { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, E    x, X    x, Y    x, Z

Proof of Theorem frgra3vlem2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-reu 2746 . . 3  |-  ( E! x  e.  { A ,  B ,  C }  { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  E! x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E ) )
2 eleq1 2519 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { A ,  B ,  C }  <->  y  e.  { A ,  B ,  C }
) )
3 preq1 4054 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  A }  =  {
y ,  A }
)
4 preq1 4054 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  B }  =  {
y ,  B }
)
53, 4preq12d 4062 . . . . . . 7  |-  ( x  =  y  ->  { {
x ,  A } ,  { x ,  B } }  =  { { y ,  A } ,  { y ,  B } } )
65sseq1d 3461 . . . . . 6  |-  ( x  =  y  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { y ,  A } ,  {
y ,  B } }  C_  ran  E ) )
72, 6anbi12d 718 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  { A ,  B ,  C }  /\  { {
x ,  A } ,  { x ,  B } }  C_  ran  E
)  <->  ( y  e. 
{ A ,  B ,  C }  /\  { { y ,  A } ,  { y ,  B } }  C_  ran  E ) ) )
87eu4 2349 . . . 4  |-  ( E! x ( x  e. 
{ A ,  B ,  C }  /\  { { x ,  A } ,  { x ,  B } }  C_  ran  E )  <->  ( E. x ( x  e. 
{ A ,  B ,  C }  /\  { { x ,  A } ,  { x ,  B } }  C_  ran  E )  /\  A. x A. y ( ( ( x  e.  { A ,  B ,  C }  /\  { {
x ,  A } ,  { x ,  B } }  C_  ran  E
)  /\  ( y  e.  { A ,  B ,  C }  /\  { { y ,  A } ,  { y ,  B } }  C_  ran  E ) )  ->  x  =  y )
) )
9 frgra3vlem1 25740 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  ->  A. x A. y ( ( ( x  e. 
{ A ,  B ,  C }  /\  { { x ,  A } ,  { x ,  B } }  C_  ran  E )  /\  (
y  e.  { A ,  B ,  C }  /\  { { y ,  A } ,  {
y ,  B } }  C_  ran  E ) )  ->  x  =  y ) )
109biantrud 510 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( E. x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  <-> 
( E. x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  /\  A. x A. y ( ( ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  /\  ( y  e. 
{ A ,  B ,  C }  /\  { { y ,  A } ,  { y ,  B } }  C_  ran  E ) )  ->  x  =  y )
) ) )
11 vex 3050 . . . . . . . . . . 11  |-  x  e. 
_V
1211eltp 4019 . . . . . . . . . 10  |-  ( x  e.  { A ,  B ,  C }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C )
)
13 preq1 4054 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
14 preq1 4054 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
1513, 14preq12d 4062 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  { {
x ,  A } ,  { x ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
1615sseq1d 3461 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
17 prex 4645 . . . . . . . . . . . . . 14  |-  { A ,  A }  e.  _V
18 prex 4645 . . . . . . . . . . . . . 14  |-  { A ,  B }  e.  _V
1917, 18prss 4129 . . . . . . . . . . . . 13  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E )
20 usgraedgrn 25120 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
21 df-ne 2626 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =/=  A  <->  -.  A  =  A )
22 eqid 2453 . . . . . . . . . . . . . . . . . . . 20  |-  A  =  A
2322pm2.24i 137 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  =  A  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2421, 23sylbi 199 . . . . . . . . . . . . . . . . . 18  |-  ( A  =/=  A  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2520, 24syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2625ex 436 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2726adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { A ,  A }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2827com12 32 . . . . . . . . . . . . . 14  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2928adantr 467 . . . . . . . . . . . . 13  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  -> 
( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
3019, 29sylbir 217 . . . . . . . . . . . 12  |-  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  ->  ( (
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
3116, 30syl6bi 232 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  ->  ( (
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
32 preq1 4054 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
33 preq1 4054 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
3432, 33preq12d 4062 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  { {
x ,  A } ,  { x ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
3534sseq1d 3461 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
36 prex 4645 . . . . . . . . . . . . . 14  |-  { B ,  A }  e.  _V
37 prex 4645 . . . . . . . . . . . . . 14  |-  { B ,  B }  e.  _V
3836, 37prss 4129 . . . . . . . . . . . . 13  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E )
39 usgraedgrn 25120 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
40 df-ne 2626 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =/=  B  <->  -.  B  =  B )
41 eqid 2453 . . . . . . . . . . . . . . . . . . . 20  |-  B  =  B
4241pm2.24i 137 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =  B  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4340, 42sylbi 199 . . . . . . . . . . . . . . . . . 18  |-  ( B  =/=  B  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4439, 43syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4544ex 436 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { B ,  B }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
4645adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { B ,  B }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
4746com12 32 . . . . . . . . . . . . . 14  |-  ( { B ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
4847adantl 468 . . . . . . . . . . . . 13  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  -> 
( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
4938, 48sylbir 217 . . . . . . . . . . . 12  |-  ( { { B ,  A } ,  { B ,  B } }  C_  ran  E  ->  ( (
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
5035, 49syl6bi 232 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  ->  ( (
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
51 preq1 4054 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  A }  =  { C ,  A }
)
52 preq1 4054 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  B }  =  { C ,  B }
)
5351, 52preq12d 4062 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  { {
x ,  A } ,  { x ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
5453sseq1d 3461 . . . . . . . . . . . 12  |-  ( x  =  C  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
55 prex 4645 . . . . . . . . . . . . . 14  |-  { C ,  A }  e.  _V
56 prex 4645 . . . . . . . . . . . . . 14  |-  { C ,  B }  e.  _V
5755, 56prss 4129 . . . . . . . . . . . . 13  |-  ( ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E )
58 ax-1 6 . . . . . . . . . . . . 13  |-  ( ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  -> 
( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
5957, 58sylbir 217 . . . . . . . . . . . 12  |-  ( { { C ,  A } ,  { C ,  B } }  C_  ran  E  ->  ( (
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
6054, 59syl6bi 232 . . . . . . . . . . 11  |-  ( x  =  C  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  ->  ( (
( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
6131, 50, 603jaoi 1333 . . . . . . . . . 10  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  ->  ( { { x ,  A } ,  {
x ,  B } }  C_  ran  E  -> 
( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
6212, 61sylbi 199 . . . . . . . . 9  |-  ( x  e.  { A ,  B ,  C }  ->  ( { { x ,  A } ,  {
x ,  B } }  C_  ran  E  -> 
( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
6362imp 431 . . . . . . . 8  |-  ( ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
6463com12 32 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( ( x  e. 
{ A ,  B ,  C }  /\  { { x ,  A } ,  { x ,  B } }  C_  ran  E )  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
6564exlimdv 1781 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( E. x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
66 prssi 4131 . . . . . . . . . . 11  |-  ( ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  ->  { { C ,  A } ,  { C ,  B } }  C_  ran  E )
6766adantl 468 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )  ->  { { C ,  A } ,  { C ,  B } }  C_  ran  E )
68673mix3d 1186 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )  ->  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  \/  { { B ,  A } ,  { B ,  B } }  C_  ran  E  \/  { { C ,  A } ,  { C ,  B } }  C_  ran  E
) )
6916, 35, 54rextpg 4026 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( E. x  e. 
{ A ,  B ,  C }  { {
x ,  A } ,  { x ,  B } }  C_  ran  E  <->  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  \/  { { B ,  A } ,  { B ,  B } }  C_  ran  E  \/  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) ) )
7069ad3antrrr 737 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )  ->  ( E. x  e.  { A ,  B ,  C }  { {
x ,  A } ,  { x ,  B } }  C_  ran  E  <->  ( { { A ,  A } ,  { A ,  B } }  C_  ran  E  \/  { { B ,  A } ,  { B ,  B } }  C_  ran  E  \/  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) ) )
7168, 70mpbird 236 . . . . . . . 8  |-  ( ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )  ->  E. x  e.  { A ,  B ,  C }  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )
72 df-rex 2745 . . . . . . . 8  |-  ( E. x  e.  { A ,  B ,  C }  { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  E. x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E ) )
7371, 72sylib 200 . . . . . . 7  |-  ( ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  /\  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )  ->  E. x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E ) )
7473ex 436 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  ->  E. x
( x  e.  { A ,  B ,  C }  /\  { {
x ,  A } ,  { x ,  B } }  C_  ran  E
) ) )
7565, 74impbid 194 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( E. x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  <-> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
7610, 75bitr3d 259 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( ( E. x
( x  e.  { A ,  B ,  C }  /\  { {
x ,  A } ,  { x ,  B } }  C_  ran  E
)  /\  A. x A. y ( ( ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  /\  ( y  e. 
{ A ,  B ,  C }  /\  { { y ,  A } ,  { y ,  B } }  C_  ran  E ) )  ->  x  =  y )
)  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
778, 76syl5bb 261 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( E! x ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  {
x ,  B } }  C_  ran  E )  <-> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
781, 77syl5bb 261 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  /\  { A ,  B ,  C } USGrph  E )  -> 
( E! x  e. 
{ A ,  B ,  C }  { {
x ,  A } ,  { x ,  B } }  C_  ran  E  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
7978ex 436 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  ( { A ,  B ,  C } USGrph  E  ->  ( E! x  e.  { A ,  B ,  C }  { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    \/ w3o 985    /\ w3a 986   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889   E!weu 2301    =/= wne 2624   E.wrex 2740   E!wreu 2741    C_ wss 3406   {cpr 3972   {ctp 3974   class class class wbr 4405   ran crn 4838   USGrph cusg 25069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523  df-usgra 25072
This theorem is referenced by:  frgra3v  25742
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