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Theorem frgra3vlem2 24979
Description: Lemma 2 for frgra3v 24980. (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 2800 . . 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 2515 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { A ,  B ,  C }  <->  y  e.  { A ,  B ,  C }
) )
3 preq1 4094 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  A }  =  {
y ,  A }
)
4 preq1 4094 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  B }  =  {
y ,  B }
)
53, 4preq12d 4102 . . . . . . 7  |-  ( x  =  y  ->  { {
x ,  A } ,  { x ,  B } }  =  { { y ,  A } ,  { y ,  B } } )
65sseq1d 3516 . . . . . 6  |-  ( x  =  y  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { y ,  A } ,  {
y ,  B } }  C_  ran  E ) )
72, 6anbi12d 710 . . . . 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 2324 . . . 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 24978 . . . . . 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 507 . . . . 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 3098 . . . . . . . . . . 11  |-  x  e. 
_V
1211eltp 4059 . . . . . . . . . 10  |-  ( x  e.  { A ,  B ,  C }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C )
)
13 preq1 4094 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
14 preq1 4094 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
1513, 14preq12d 4102 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  { {
x ,  A } ,  { x ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
1615sseq1d 3516 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
17 prex 4679 . . . . . . . . . . . . . 14  |-  { A ,  A }  e.  _V
18 prex 4679 . . . . . . . . . . . . . 14  |-  { A ,  B }  e.  _V
1917, 18prss 4169 . . . . . . . . . . . . 13  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E )
20 usgraedgrn 24359 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
21 df-ne 2640 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =/=  A  <->  -.  A  =  A )
22 eqid 2443 . . . . . . . . . . . . . . . . . . . 20  |-  A  =  A
2322pm2.24i 144 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  A  =  A  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2421, 23sylbi 195 . . . . . . . . . . . . . . . . . 18  |-  ( A  =/=  A  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2520, 24syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2625ex 434 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2726adantl 466 . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 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 228 . . . . . . . . . . 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 4094 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
33 preq1 4094 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
3432, 33preq12d 4102 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  { {
x ,  A } ,  { x ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
3534sseq1d 3516 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
36 prex 4679 . . . . . . . . . . . . . 14  |-  { B ,  A }  e.  _V
37 prex 4679 . . . . . . . . . . . . . 14  |-  { B ,  B }  e.  _V
3836, 37prss 4169 . . . . . . . . . . . . 13  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E )
39 usgraedgrn 24359 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
40 df-ne 2640 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =/=  B  <->  -.  B  =  B )
41 eqid 2443 . . . . . . . . . . . . . . . . . . . 20  |-  B  =  B
4241pm2.24i 144 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =  B  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4340, 42sylbi 195 . . . . . . . . . . . . . . . . . 18  |-  ( B  =/=  B  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4439, 43syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  -> 
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
4544ex 434 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { B ,  B }  e.  ran  E  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
4645adantl 466 . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 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 228 . . . . . . . . . . 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 4094 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  A }  =  { C ,  A }
)
52 preq1 4094 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  B }  =  { C ,  B }
)
5351, 52preq12d 4102 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  { {
x ,  A } ,  { x ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
5453sseq1d 3516 . . . . . . . . . . . 12  |-  ( x  =  C  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
55 prex 4679 . . . . . . . . . . . . . 14  |-  { C ,  A }  e.  _V
56 prex 4679 . . . . . . . . . . . . . 14  |-  { C ,  B }  e.  _V
5755, 56prss 4169 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 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 228 . . . . . . . . . . 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 1292 . . . . . . . . . 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 195 . . . . . . . . 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 429 . . . . . . . 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 31 . . . . . . 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 1711 . . . . . 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 4171 . . . . . . . . . . 11  |-  ( ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  ->  { { C ,  A } ,  { C ,  B } }  C_  ran  E )
6766adantl 466 . . . . . . . . . 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 1174 . . . . . . . . 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 4066 . . . . . . . . . 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 729 . . . . . . . . 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 232 . . . . . . . 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 2799 . . . . . . . 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 196 . . . . . . 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 434 . . . . . 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 191 . . . . 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 255 . . . 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 257 . . 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 257 . 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 434 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 184    /\ wa 369    \/ w3o 973    /\ w3a 974   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   E!weu 2268    =/= wne 2638   E.wrex 2794   E!wreu 2795    C_ wss 3461   {cpr 4016   {ctp 4018   class class class wbr 4437   ran crn 4990   USGrph cusg 24308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388  df-usgra 24311
This theorem is referenced by:  frgra3v  24980
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