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Theorem frgra3vlem2 25571
Description: Lemma 2 for frgra3v 25572. (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 2780 . . 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 2492 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { A ,  B ,  C }  <->  y  e.  { A ,  B ,  C }
) )
3 preq1 4073 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  A }  =  {
y ,  A }
)
4 preq1 4073 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  B }  =  {
y ,  B }
)
53, 4preq12d 4081 . . . . . . 7  |-  ( x  =  y  ->  { {
x ,  A } ,  { x ,  B } }  =  { { y ,  A } ,  { y ,  B } } )
65sseq1d 3488 . . . . . 6  |-  ( x  =  y  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { y ,  A } ,  {
y ,  B } }  C_  ran  E ) )
72, 6anbi12d 715 . . . . 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 2312 . . . 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 25570 . . . . . 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 509 . . . . 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 3081 . . . . . . . . . . 11  |-  x  e. 
_V
1211eltp 4039 . . . . . . . . . 10  |-  ( x  e.  { A ,  B ,  C }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C )
)
13 preq1 4073 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
14 preq1 4073 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
1513, 14preq12d 4081 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  { {
x ,  A } ,  { x ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
1615sseq1d 3488 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
17 prex 4655 . . . . . . . . . . . . . 14  |-  { A ,  A }  e.  _V
18 prex 4655 . . . . . . . . . . . . . 14  |-  { A ,  B }  e.  _V
1917, 18prss 4148 . . . . . . . . . . . . 13  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E )
20 usgraedgrn 24951 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
21 df-ne 2618 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =/=  A  <->  -.  A  =  A )
22 eqid 2420 . . . . . . . . . . . . . . . . . . . 20  |-  A  =  A
2322pm2.24i 136 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  =  A  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2421, 23sylbi 198 . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2726adantl 467 . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 216 . . . . . . . . . . . 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 231 . . . . . . . . . . 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 4073 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
33 preq1 4073 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
3432, 33preq12d 4081 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  { {
x ,  A } ,  { x ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
3534sseq1d 3488 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
36 prex 4655 . . . . . . . . . . . . . 14  |-  { B ,  A }  e.  _V
37 prex 4655 . . . . . . . . . . . . . 14  |-  { B ,  B }  e.  _V
3836, 37prss 4148 . . . . . . . . . . . . 13  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E )
39 usgraedgrn 24951 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
40 df-ne 2618 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =/=  B  <->  -.  B  =  B )
41 eqid 2420 . . . . . . . . . . . . . . . . . . . 20  |-  B  =  B
4241pm2.24i 136 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =  B  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4340, 42sylbi 198 . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { B ,  B }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
4645adantl 467 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 216 . . . . . . . . . . . 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 231 . . . . . . . . . . 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 4073 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  A }  =  { C ,  A }
)
52 preq1 4073 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  B }  =  { C ,  B }
)
5351, 52preq12d 4081 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  { {
x ,  A } ,  { x ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
5453sseq1d 3488 . . . . . . . . . . . 12  |-  ( x  =  C  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
55 prex 4655 . . . . . . . . . . . . . 14  |-  { C ,  A }  e.  _V
56 prex 4655 . . . . . . . . . . . . . 14  |-  { C ,  B }  e.  _V
5755, 56prss 4148 . . . . . . . . . . . . 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 216 . . . . . . . . . . . 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 231 . . . . . . . . . . 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 1327 . . . . . . . . . 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 198 . . . . . . . . 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 430 . . . . . . . 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 1768 . . . . . 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 4150 . . . . . . . . . . 11  |-  ( ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  ->  { { C ,  A } ,  { C ,  B } }  C_  ran  E )
6766adantl 467 . . . . . . . . . 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 1182 . . . . . . . . 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 4046 . . . . . . . . . 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 734 . . . . . . . . 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 235 . . . . . . . 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 2779 . . . . . . . 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 199 . . . . . . 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 435 . . . . . 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 193 . . . . 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 258 . . . 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 260 . . 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 260 . 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 435 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 187    /\ wa 370    \/ w3o 981    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1867   E!weu 2263    =/= wne 2616   E.wrex 2774   E!wreu 2775    C_ wss 3433   {cpr 3995   {ctp 3997   class class class wbr 4417   ran crn 4846   USGrph cusg 24900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-hash 12502  df-usgra 24903
This theorem is referenced by:  frgra3v  25572
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