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Theorem nb3graprlem1 25172
Description: Lemma 1 for nb3grapr 25174. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
Assertion
Ref Expression
nb3graprlem1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )

Proof of Theorem nb3graprlem1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 prid1g 4077 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
213ad2ant2 1029 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  B  e.  { B ,  C } )
32adantr 467 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { B ,  C } )
43adantr 467 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  B  e.  { B ,  C }
)
5 eleq2 2517 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
65eqcoms 2458 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
76adantl 468 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( B  e.  { B ,  C } 
<->  B  e.  ( <. V ,  E >. Neighbors  A
) ) )
84, 7mpbid 214 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  B  e.  ( <. V ,  E >. Neighbors  A ) )
9 nbgraeledg 25151 . . . . . . 7  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { B ,  A }  e.  ran  E ) )
10 prcom 4049 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1110a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { B ,  A }  =  { A ,  B }
)
1211eleq1d 2512 . . . . . . 7  |-  ( V USGrph  E  ->  ( { B ,  A }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
139, 12bitrd 257 . . . . . 6  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  B }  e.  ran  E ) )
1413adantl 468 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( B  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  B }  e.  ran  E ) )
1514ad2antlr 732 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  B }  e.  ran  E ) )
168, 15mpbid 214 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  { A ,  B }  e.  ran  E )
17 prid2g 4078 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
18173ad2ant3 1030 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { B ,  C } )
1918adantr 467 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { B ,  C } )
2019adantr 467 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  C  e.  { B ,  C }
)
21 eleq2 2517 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2221eqcoms 2458 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2322adantl 468 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( C  e.  { B ,  C } 
<->  C  e.  ( <. V ,  E >. Neighbors  A
) ) )
2420, 23mpbid 214 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  C  e.  ( <. V ,  E >. Neighbors  A ) )
25 nbgraeledg 25151 . . . . . . 7  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { C ,  A }  e.  ran  E ) )
26 prcom 4049 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
2726a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { C ,  A }  =  { A ,  C }
)
2827eleq1d 2512 . . . . . . 7  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E ) )
2925, 28bitrd 257 . . . . . 6  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  C }  e.  ran  E ) )
3029adantl 468 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( C  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  C }  e.  ran  E ) )
3130ad2antlr 732 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  C }  e.  ran  E ) )
3224, 31mpbid 214 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  { A ,  C }  e.  ran  E )
3316, 32jca 535 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )
34 nbusgra 25149 . . . . 5  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  A )  =  {
v  e.  V  |  { A ,  v }  e.  ran  E }
)
3534ad2antll 734 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( <. V ,  E >. Neighbors  A
)  =  { v  e.  V  |  { A ,  v }  e.  ran  E } )
3635adantr 467 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  A
)  =  { v  e.  V  |  { A ,  v }  e.  ran  E } )
37 eleq2 2517 . . . . . . . . . 10  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  V  <->  v  e.  { A ,  B ,  C }
) )
3837ad2antrl 733 . . . . . . . . 9  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  e.  V  <->  v  e.  { A ,  B ,  C } ) )
3938adantr 467 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  <->  v  e.  { A ,  B ,  C } ) )
40 vex 3047 . . . . . . . . . . 11  |-  v  e. 
_V
4140eltp 4016 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
42 usgraedgrn 25101 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  A  =/=  v )
43 df-ne 2623 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
44 pm2.24 113 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =  v  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4544eqcoms 2458 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  A  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4645com12 32 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  =  v  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) )
4743, 46sylbi 199 . . . . . . . . . . . . . . . 16  |-  ( A  =/=  v  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
4842, 47syl 17 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  (
v  =  A  -> 
( v  =  B  \/  v  =  C ) ) )
4948ex 436 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5049ad2antll 734 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5150adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5251com3r 82 . . . . . . . . . . 11  |-  ( v  =  A  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) ) )
53 orc 387 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  (
v  =  B  \/  v  =  C )
)
5453a1d 26 . . . . . . . . . . . 12  |-  ( v  =  B  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) )
5554a1d 26 . . . . . . . . . . 11  |-  ( v  =  B  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) ) )
56 olc 386 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  (
v  =  B  \/  v  =  C )
)
5756a1d 26 . . . . . . . . . . . 12  |-  ( v  =  C  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) )
5857a1d 26 . . . . . . . . . . 11  |-  ( v  =  C  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  -> 
( v  =  B  \/  v  =  C ) ) ) )
5952, 55, 583jaoi 1330 . . . . . . . . . 10  |-  ( ( v  =  A  \/  v  =  B  \/  v  =  C )  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E
) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6041, 59sylbi 199 . . . . . . . . 9  |-  ( v  e.  { A ,  B ,  C }  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E
) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6160com12 32 . . . . . . . 8  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  { A ,  B ,  C }  ->  ( { A , 
v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6239, 61sylbid 219 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  -> 
( { A , 
v }  e.  ran  E  ->  ( v  =  B  \/  v  =  C ) ) ) )
6362impd 433 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  ran  E
)  ->  ( v  =  B  \/  v  =  C ) ) )
64 eqid 2450 . . . . . . . . . . . . . . . . . . . . 21  |-  B  =  B
65643mix2i 1180 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
66 eltpg 4013 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
6765, 66mpbiri 237 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  Y  ->  B  e.  { A ,  B ,  C } )
6867a1d 26 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  Y  ->  (
( V  =  { A ,  B ,  C }  /\  V USGrph  E
)  ->  B  e.  { A ,  B ,  C } ) )
69683ad2ant2 1029 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  B  e.  { A ,  B ,  C } ) )
7069imp 431 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { A ,  B ,  C } )
7170adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  B  e.  { A ,  B ,  C } )
72 eleq1 2516 . . . . . . . . . . . . . . . . 17  |-  ( v  =  B  ->  (
v  e.  { A ,  B ,  C }  <->  B  e.  { A ,  B ,  C }
) )
7372bicomd 205 . . . . . . . . . . . . . . . 16  |-  ( v  =  B  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7473adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C } ) )
7571, 74mpbid 214 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  v  e.  { A ,  B ,  C } )
7637bicomd 205 . . . . . . . . . . . . . . . 16  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V
) )
7776ad2antrl 733 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
7877adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
7975, 78mpbid 214 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  v  e.  V )
8079ex 436 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  B  -> 
v  e.  V ) )
8180adantr 467 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  B  -> 
v  e.  V ) )
8281impcom 432 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
v  e.  V )
83 preq2 4051 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8483eleq1d 2512 . . . . . . . . . . . . . 14  |-  ( B  =  v  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8584eqcoms 2458 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8685biimpcd 228 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  ran  E  ->  (
v  =  B  ->  { A ,  v }  e.  ran  E ) )
8786ad2antrl 733 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  B  ->  { A ,  v }  e.  ran  E ) )
8887impcom 432 . . . . . . . . . 10  |-  ( ( v  =  B  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  ->  { A ,  v }  e.  ran  E )
8982, 88jca 535 . . . . . . . . 9  |-  ( ( v  =  B  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  ran  E
) )
9089ex 436 . . . . . . . 8  |-  ( v  =  B  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
91 tpid3g 4086 . . . . . . . . . . . . . . . . . . 19  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
9291a1d 26 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  Z  ->  (
( V  =  { A ,  B ,  C }  /\  V USGrph  E
)  ->  C  e.  { A ,  B ,  C } ) )
93923ad2ant3 1030 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  C  e.  { A ,  B ,  C } ) )
9493imp 431 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { A ,  B ,  C } )
9594adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  C  e.  { A ,  B ,  C } )
96 eleq1 2516 . . . . . . . . . . . . . . . . 17  |-  ( v  =  C  ->  (
v  e.  { A ,  B ,  C }  <->  C  e.  { A ,  B ,  C }
) )
9796bicomd 205 . . . . . . . . . . . . . . . 16  |-  ( v  =  C  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
9897adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C } ) )
9995, 98mpbid 214 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  v  e.  { A ,  B ,  C } )
10077adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V ) )
10199, 100mpbid 214 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  v  e.  V )
102101ex 436 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  C  -> 
v  e.  V ) )
103102adantr 467 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  C  -> 
v  e.  V ) )
104103impcom 432 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
v  e.  V )
105 preq2 4051 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
106105eleq1d 2512 . . . . . . . . . . . . . 14  |-  ( C  =  v  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
107106eqcoms 2458 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
108107biimpcd 228 . . . . . . . . . . . 12  |-  ( { A ,  C }  e.  ran  E  ->  (
v  =  C  ->  { A ,  v }  e.  ran  E ) )
109108ad2antll 734 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  =  C  ->  { A ,  v }  e.  ran  E ) )
110109impcom 432 . . . . . . . . . 10  |-  ( ( v  =  C  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  ->  { A ,  v }  e.  ran  E )
111104, 110jca 535 . . . . . . . . 9  |-  ( ( v  =  C  /\  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )  -> 
( v  e.  V  /\  { A ,  v }  e.  ran  E
) )
112111ex 436 . . . . . . . 8  |-  ( v  =  C  ->  (
( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
11390, 112jaoi 381 . . . . . . 7  |-  ( ( v  =  B  \/  v  =  C )  ->  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E
) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
114113com12 32 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
( v  =  B  \/  v  =  C )  ->  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) ) )
11563, 114impbid 194 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  (
( v  e.  V  /\  { A ,  v }  e.  ran  E
)  <->  ( v  =  B  \/  v  =  C ) ) )
116115abbidv 2568 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }  =  { v  |  ( v  =  B  \/  v  =  C ) } )
117 df-rab 2745 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }
118 dfpr2 3982 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
119116, 117, 1183eqtr4g 2509 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { B ,  C }
)
12036, 119eqtrd 2484 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  A
)  =  { B ,  C } )
12133, 120impbida 842 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    \/ w3o 983    /\ w3a 984    = wceq 1443    e. wcel 1886   {cab 2436    =/= wne 2621   {crab 2740   {cpr 3969   {ctp 3971   <.cop 3973   class class class wbr 4401   ran crn 4834  (class class class)co 6288   USGrph cusg 25050   Neighbors cnbgra 25138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-hash 12513  df-usgra 25053  df-nbgra 25141
This theorem is referenced by:  nb3grapr  25174  nb3grapr2  25175  nb3gra2nb  25176
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