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Theorem frgra3vlem2 30598
Description: Lemma 2 for frgra3v 30599. (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 2727 . . 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 2503 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { A ,  B ,  C }  <->  y  e.  { A ,  B ,  C }
) )
3 preq1 3959 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  A }  =  {
y ,  A }
)
4 preq1 3959 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  B }  =  {
y ,  B }
)
53, 4preq12d 3967 . . . . . . 7  |-  ( x  =  y  ->  { {
x ,  A } ,  { x ,  B } }  =  { { y ,  A } ,  { y ,  B } } )
65sseq1d 3388 . . . . . 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 2318 . . . 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 30597 . . . . . 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 2980 . . . . . . . . . . 11  |-  x  e. 
_V
1211eltp 3926 . . . . . . . . . 10  |-  ( x  e.  { A ,  B ,  C }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C )
)
13 preq1 3959 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  A }  =  { A ,  A }
)
14 preq1 3959 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
1513, 14preq12d 3967 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  { {
x ,  A } ,  { x ,  B } }  =  { { A ,  A } ,  { A ,  B } } )
1615sseq1d 3388 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E ) )
17 prex 4539 . . . . . . . . . . . . . 14  |-  { A ,  A }  e.  _V
18 prex 4539 . . . . . . . . . . . . . 14  |-  { A ,  B }  e.  _V
1917, 18prss 4032 . . . . . . . . . . . . 13  |-  ( ( { A ,  A }  e.  ran  E  /\  { A ,  B }  e.  ran  E )  <->  { { A ,  A } ,  { A ,  B } }  C_  ran  E )
20 usgraedgrn 23305 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
21 df-ne 2613 . . . . . . . . . . . . . . . . . . 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 3959 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  A }  =  { B ,  A }
)
33 preq1 3959 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  { x ,  B }  =  { B ,  B }
)
3432, 33preq12d 3967 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  { {
x ,  A } ,  { x ,  B } }  =  { { B ,  A } ,  { B ,  B } } )
3534sseq1d 3388 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E ) )
36 prex 4539 . . . . . . . . . . . . . 14  |-  { B ,  A }  e.  _V
37 prex 4539 . . . . . . . . . . . . . 14  |-  { B ,  B }  e.  _V
3836, 37prss 4032 . . . . . . . . . . . . 13  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  B }  e.  ran  E )  <->  { { B ,  A } ,  { B ,  B } }  C_  ran  E )
39 usgraedgrn 23305 . . . . . . . . . . . . . . . . . 18  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { B ,  B }  e.  ran  E )  ->  B  =/=  B )
40 df-ne 2613 . . . . . . . . . . . . . . . . . . 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 3959 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  A }  =  { C ,  A }
)
52 preq1 3959 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  { x ,  B }  =  { C ,  B }
)
5351, 52preq12d 3967 . . . . . . . . . . . . 13  |-  ( x  =  C  ->  { {
x ,  A } ,  { x ,  B } }  =  { { C ,  A } ,  { C ,  B } } )
5453sseq1d 3388 . . . . . . . . . . . 12  |-  ( x  =  C  ->  ( { { x ,  A } ,  { x ,  B } }  C_  ran  E  <->  { { C ,  A } ,  { C ,  B } }  C_  ran  E ) )
55 prex 4539 . . . . . . . . . . . . . 14  |-  { C ,  A }  e.  _V
56 prex 4539 . . . . . . . . . . . . . 14  |-  { C ,  B }  e.  _V
5755, 56prss 4032 . . . . . . . . . . . . 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 1281 . . . . . . . . . 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 1690 . . . . . 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 4034 . . . . . . . . . . 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 )
68 3mix3 1159 . . . . . . . . . 10  |-  ( { { C ,  A } ,  { C ,  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
) )
6967, 68syl 16 . . . . . . . . 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
) )
7016, 35, 54rextpg 3933 . . . . . . . . . 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 ) ) )
7170ad3antrrr 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 ) ) )
7269, 71mpbird 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 )
73 df-rex 2726 . . . . . . . 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 ) )
7472, 73sylib 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 ) )
7574ex 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
) ) )
7665, 75impbid 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 ) ) )
7710, 76bitr3d 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 ) ) )
788, 77syl5bb 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 ) ) )
791, 78syl5bb 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 ) ) )
8079ex 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 964    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253    =/= wne 2611   E.wrex 2721   E!wreu 2722    C_ wss 3333   {cpr 3884   {ctp 3886   class class class wbr 4297   ran crn 4846   USGrph cusg 23269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-hash 12109  df-usgra 23271
This theorem is referenced by:  frgra3v  30599
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