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Theorem nb3graprlem1 25065
Description: Lemma 1 for nb3grapr 25067. (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 4100 . . . . . . . 8  |-  ( B  e.  Y  ->  B  e.  { B ,  C } )
213ad2ant2 1027 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  B  e.  { B ,  C } )
32adantr 466 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { B ,  C } )
43adantr 466 . . . . 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 2493 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
65eqcoms 2432 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  ->  ( B  e.  { B ,  C }  <->  B  e.  ( <. V ,  E >. Neighbors  A ) ) )
76adantl 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 } 
<->  B  e.  ( <. V ,  E >. Neighbors  A
) ) )
84, 7mpbid 213 . . . 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 25044 . . . . . . 7  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { B ,  A }  e.  ran  E ) )
10 prcom 4072 . . . . . . . . 9  |-  { B ,  A }  =  { A ,  B }
1110a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { B ,  A }  =  { A ,  B }
)
1211eleq1d 2489 . . . . . . 7  |-  ( V USGrph  E  ->  ( { B ,  A }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
139, 12bitrd 256 . . . . . 6  |-  ( V USGrph  E  ->  ( B  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  B }  e.  ran  E ) )
1413adantl 467 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( B  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  B }  e.  ran  E ) )
1514ad2antlr 731 . . . 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 213 . . 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 4101 . . . . . . . 8  |-  ( C  e.  Z  ->  C  e.  { B ,  C } )
18173ad2ant3 1028 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  { B ,  C } )
1918adantr 466 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { B ,  C } )
2019adantr 466 . . . . 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 2493 . . . . . . 7  |-  ( { B ,  C }  =  ( <. V ,  E >. Neighbors  A )  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2221eqcoms 2432 . . . . . 6  |-  ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  ->  ( C  e.  { B ,  C }  <->  C  e.  ( <. V ,  E >. Neighbors  A ) ) )
2322adantl 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 } 
<->  C  e.  ( <. V ,  E >. Neighbors  A
) ) )
2420, 23mpbid 213 . . . 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 25044 . . . . . . 7  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { C ,  A }  e.  ran  E ) )
26 prcom 4072 . . . . . . . . 9  |-  { C ,  A }  =  { A ,  C }
2726a1i 11 . . . . . . . 8  |-  ( V USGrph  E  ->  { C ,  A }  =  { A ,  C }
)
2827eleq1d 2489 . . . . . . 7  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E ) )
2925, 28bitrd 256 . . . . . 6  |-  ( V USGrph  E  ->  ( C  e.  ( <. V ,  E >. Neighbors  A )  <->  { A ,  C }  e.  ran  E ) )
3029adantl 467 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( C  e.  (
<. V ,  E >. Neighbors  A
)  <->  { A ,  C }  e.  ran  E ) )
3130ad2antlr 731 . . . 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 213 . . 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 534 . 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 25042 . . . . 5  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  A )  =  {
v  e.  V  |  { A ,  v }  e.  ran  E }
)
3534ad2antll 733 . . . 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 466 . . 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 2493 . . . . . . . . . 10  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  V  <->  v  e.  { A ,  B ,  C }
) )
3837ad2antrl 732 . . . . . . . . 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 466 . . . . . . . 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 3081 . . . . . . . . . . 11  |-  v  e. 
_V
4140eltp 4039 . . . . . . . . . 10  |-  ( v  e.  { A ,  B ,  C }  <->  ( v  =  A  \/  v  =  B  \/  v  =  C )
)
42 usgraedgrn 24995 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  { A ,  v }  e.  ran  E )  ->  A  =/=  v )
43 df-ne 2618 . . . . . . . . . . . . . . . . 17  |-  ( A  =/=  v  <->  -.  A  =  v )
44 pm2.24 112 . . . . . . . . . . . . . . . . . . 19  |-  ( A  =  v  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4544eqcoms 2432 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  A  ->  ( -.  A  =  v  ->  ( v  =  B  \/  v  =  C ) ) )
4645com12 32 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  =  v  -> 
( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) )
4743, 46sylbi 198 . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( { A ,  v }  e.  ran  E  ->  ( v  =  A  ->  ( v  =  B  \/  v  =  C ) ) ) )
5049ad2antll 733 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 386 . . . . . . . . . . . . 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 385 . . . . . . . . . . . . 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 1327 . . . . . . . . . 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 198 . . . . . . . . 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 218 . . . . . . 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 432 . . . . . 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 2420 . . . . . . . . . . . . . . . . . . . . 21  |-  B  =  B
65643mix2i 1178 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
66 eltpg 4036 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
6765, 66mpbiri 236 . . . . . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  B  e.  { A ,  B ,  C } ) )
7069imp 430 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  B  e.  { A ,  B ,  C } )
7170adantr 466 . . . . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . . . . 17  |-  ( v  =  B  ->  (
v  e.  { A ,  B ,  C }  <->  B  e.  { A ,  B ,  C }
) )
7372bicomd 204 . . . . . . . . . . . . . . . 16  |-  ( v  =  B  ->  ( B  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
7473adantl 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 }  <->  v  e.  { A ,  B ,  C } ) )
7571, 74mpbid 213 . . . . . . . . . . . . . 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 204 . . . . . . . . . . . . . . . 16  |-  ( V  =  { A ,  B ,  C }  ->  ( v  e.  { A ,  B ,  C }  <->  v  e.  V
) )
7776ad2antrl 732 . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  B )  ->  v  e.  V )
8079ex 435 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  B  -> 
v  e.  V ) )
8180adantr 466 . . . . . . . . . . 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 431 . . . . . . . . . 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 4074 . . . . . . . . . . . . . . 15  |-  ( B  =  v  ->  { A ,  B }  =  { A ,  v }
)
8483eleq1d 2489 . . . . . . . . . . . . . 14  |-  ( B  =  v  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8584eqcoms 2432 . . . . . . . . . . . . 13  |-  ( v  =  B  ->  ( { A ,  B }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
8685biimpcd 227 . . . . . . . . . . . 12  |-  ( { A ,  B }  e.  ran  E  ->  (
v  =  B  ->  { A ,  v }  e.  ran  E ) )
8786ad2antrl 732 . . . . . . . . . . 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 431 . . . . . . . . . 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 534 . . . . . . . . 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 435 . . . . . . . 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 4109 . . . . . . . . . . . . . . . . . . 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 1028 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  ->  C  e.  { A ,  B ,  C } ) )
9493imp 430 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  C  e.  { A ,  B ,  C } )
9594adantr 466 . . . . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . . . . 17  |-  ( v  =  C  ->  (
v  e.  { A ,  B ,  C }  <->  C  e.  { A ,  B ,  C }
) )
9796bicomd 204 . . . . . . . . . . . . . . . 16  |-  ( v  =  C  ->  ( C  e.  { A ,  B ,  C }  <->  v  e.  { A ,  B ,  C }
) )
9897adantl 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 }  <->  v  e.  { A ,  B ,  C } ) )
9995, 98mpbid 213 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  /\  v  =  C )  ->  v  e.  V )
102101ex 435 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
v  =  C  -> 
v  e.  V ) )
103102adantr 466 . . . . . . . . . . 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 431 . . . . . . . . . 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 4074 . . . . . . . . . . . . . . 15  |-  ( C  =  v  ->  { A ,  C }  =  { A ,  v }
)
106105eleq1d 2489 . . . . . . . . . . . . . 14  |-  ( C  =  v  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
107106eqcoms 2432 . . . . . . . . . . . . 13  |-  ( v  =  C  ->  ( { A ,  C }  e.  ran  E  <->  { A ,  v }  e.  ran  E ) )
108107biimpcd 227 . . . . . . . . . . . 12  |-  ( { A ,  C }  e.  ran  E  ->  (
v  =  C  ->  { A ,  v }  e.  ran  E ) )
109108ad2antll 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  =  C  ->  { A ,  v }  e.  ran  E ) )
110109impcom 431 . . . . . . . . . 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 534 . . . . . . . . 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 435 . . . . . . . 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 380 . . . . . . 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 193 . . . . 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 2556 . . . 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 2782 . . . 4  |-  { v  e.  V  |  { A ,  v }  e.  ran  E }  =  { v  |  ( v  e.  V  /\  { A ,  v }  e.  ran  E ) }
118 dfpr2 4008 . . . 4  |-  { B ,  C }  =  {
v  |  ( v  =  B  \/  v  =  C ) }
119116, 117, 1183eqtr4g 2486 . . 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 2461 . 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 840 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 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1867   {cab 2405    =/= wne 2616   {crab 2777   {cpr 3995   {ctp 3997   <.cop 3999   class class class wbr 4417   ran crn 4846  (class class class)co 6296   USGrph cusg 24944   Neighbors cnbgra 25031
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 24947  df-nbgra 25034
This theorem is referenced by:  nb3grapr  25067  nb3grapr2  25068  nb3gra2nb  25069
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