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Theorem usg2wlkonot 25010
Description: A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to proof this the cases  A  =  B and/or  B  =  C must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
usg2wlkonot  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )

Proof of Theorem usg2wlkonot
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 24465 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2wlkonotot 25000 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
323adantr2 1156 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
41, 3sylan 471 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
5 vex 3112 . . . . . . . . . . . 12  |-  f  e. 
_V
6 vex 3112 . . . . . . . . . . . 12  |-  p  e. 
_V
75, 6pm3.2i 455 . . . . . . . . . . 11  |-  ( f  e.  _V  /\  p  e.  _V )
8 is2wlk 24694 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
91, 7, 8sylancl 662 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
10 preq12 4113 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
11103adant3 1016 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
1211eqeq2d 2471 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B } 
<->  ( E `  (
f `  0 )
)  =  { ( p `  0 ) ,  ( p ` 
1 ) } ) )
13 preq12 4113 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =  ( p `
 1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
14133adant1 1014 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
1514eqeq2d 2471 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
1 ) )  =  { B ,  C } 
<->  ( E `  (
f `  1 )
)  =  { ( p `  1 ) ,  ( p ` 
2 ) } ) )
1612, 15anbi12d 710 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( ( E `  ( f `
 0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( f `  0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } ) ) )
1716bicomd 201 . . . . . . . . . . . . 13  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( ( E `  ( f `
 0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } )  <->  ( ( E `  ( f `  0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } ) ) )
18173anbi3d 1305 . . . . . . . . . . . 12  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } ) ) ) )
19 usgrafun 24476 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  Fun  E )
20 c0ex 9607 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  0  e.  _V
2120prid1 4140 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  { 0 ,  1 }
22 fzo0to2pr 11902 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2321, 22eleqtrri 2544 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ( 0..^ 2 )
24 ffvelrn 6030 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( f ` 
0 )  e.  dom  E )
2523, 24mpan2 671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  0 )  e.  dom  E )
26 fvelrn 6025 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  0 )  e.  dom  E )  -> 
( E `  (
f `  0 )
)  e.  ran  E
)
2725, 26sylan2 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  0 )
)  e.  ran  E
)
28 eleq1 2529 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 0 ) )  =  { A ,  B }  ->  ( ( E `  ( f `
 0 ) )  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
2927, 28syl5ibcom 220 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  0
) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
30 1ex 9608 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  _V
3130prid2 4141 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  e.  { 0 ,  1 }
3231, 22eleqtrri 2544 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  ( 0..^ 2 )
33 ffvelrn 6030 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( f ` 
1 )  e.  dom  E )
3432, 33mpan2 671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  1 )  e.  dom  E )
35 fvelrn 6025 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  1 )  e.  dom  E )  -> 
( E `  (
f `  1 )
)  e.  ran  E
)
3634, 35sylan2 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  1 )
)  e.  ran  E
)
37 eleq1 2529 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 1 ) )  =  { B ,  C }  ->  ( ( E `  ( f `
 1 ) )  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
3836, 37syl5ibcom 220 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  1
) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
3929, 38anim12d 563 . . . . . . . . . . . . . . . . . 18  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4019, 39sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4140a1d 25 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
4241expcom 435 . . . . . . . . . . . . . . 15  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4342com24 87 . . . . . . . . . . . . . 14  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( ( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } )  -> 
( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4443a1d 25 . . . . . . . . . . . . 13  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( p : ( 0 ... 2 ) --> V  ->  ( ( ( E `  ( f `
 0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) ) )
45443imp 1190 . . . . . . . . . . . 12  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
4618, 45syl6bi 228 . . . . . . . . . . 11  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4746com14 88 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
489, 47sylbid 215 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4948com14 88 . . . . . . . 8  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5049expdcom 439 . . . . . . 7  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) ) )
51503imp 1190 . . . . . 6  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5251com13 80 . . . . 5  |-  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5352imp 429 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5453exlimdvv 1726 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
55 usg2wlk 24744 . . . . 5  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
56553expib 1199 . . . 4  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
5756adantr 465 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
5854, 57impbid 191 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
594, 58bitrd 253 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   _Vcvv 3109   {cpr 4034   <.cotp 4040   class class class wbr 4456   dom cdm 5008   ran crn 5009   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   2c2 10606   ...cfz 11697  ..^cfzo 11821   #chash 12408   USGrph cusg 24457   Walks cwalk 24625   2WalksOnOt c2wlkonot 24982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-usgra 24460  df-wlk 24635  df-wlkon 24641  df-2wlkonot 24985
This theorem is referenced by:  usg2spthonot  25015  usg2spthonot0  25016  frg2woteu  25182  frg2woteqm  25186
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