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Theorem constr3trllem5 24358
Description: Lemma for constr3trl 24363. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
constr3cycl.p  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
Assertion
Ref Expression
constr3trllem5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
Distinct variable groups:    k, E    k, F    P, k
Allowed substitution hints:    A( k)    B( k)    C( k)    V( k)

Proof of Theorem constr3trllem5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgraedgrnv 24081 . . . . . . 7  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
21ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  ( A  e.  V  /\  B  e.  V ) ) )
3 usgraedgrnv 24081 . . . . . . 7  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
43ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  ( B  e.  V  /\  C  e.  V ) ) )
5 usgraedgrnv 24081 . . . . . . 7  |-  ( ( V USGrph  E  /\  { C ,  A }  e.  ran  E )  ->  ( C  e.  V  /\  A  e.  V ) )
65ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  ->  ( C  e.  V  /\  A  e.  V ) ) )
72, 4, 63anim123d 1306 . . . . 5  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )  /\  ( C  e.  V  /\  A  e.  V
) ) ) )
87imp 429 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )  /\  ( C  e.  V  /\  A  e.  V
) ) )
9 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
10 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  B  e.  V )
11 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
12 constr3cycl.f . . . . . . . 8  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
13 constr3cycl.p . . . . . . . 8  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
1412, 13constr3lem4 24351 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) ) )
159, 10, 11, 14syl3anc 1228 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) ) )
16 usgraf 24050 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
17 f1f1orn 5827 . . . . . . . . . . . . 13  |-  ( E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  ->  E : dom  E -1-1-onto-> ran  E )
18 f1ocnvfv2 6171 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { A ,  B }  e.  ran  E )  ->  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }
)
1918ex 434 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B } ) )
20 f1ocnvfv2 6171 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( E `  ( `' E `  { B ,  C }
) )  =  { B ,  C }
)
2120ex 434 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { B ,  C }  e.  ran  E  ->  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
22 f1ocnvfv2 6171 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { C ,  A }  e.  ran  E )  ->  ( E `  ( `' E `  { C ,  A }
) )  =  { C ,  A }
)
2322ex 434 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { C ,  A }  e.  ran  E  ->  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2419, 21, 233anim123d 1306 . . . . . . . . . . . . 13  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) ) )
2516, 17, 243syl 20 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) ) )
2625imp 429 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2726adantl 466 . . . . . . . . . 10  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2812, 13constr3lem5 24352 . . . . . . . . . . 11  |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )
29 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( `' E `  { A ,  B } ) ) )
30293ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  0 )
)  =  ( E `
 ( `' E `  { A ,  B } ) ) )
3130eqeq1d 2469 . . . . . . . . . . . 12  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( E `  ( F `  0 ) )  =  { A ,  B }  <->  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }
) )
32 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( ( F `  1 )  =  ( `' E `  { B ,  C } )  ->  ( E `  ( F `  1 ) )  =  ( E `  ( `' E `  { B ,  C } ) ) )
33323ad2ant2 1018 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  1 )
)  =  ( E `
 ( `' E `  { B ,  C } ) ) )
3433eqeq1d 2469 . . . . . . . . . . . 12  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( E `  ( F `  1 ) )  =  { B ,  C }  <->  ( E `  ( `' E `  { B ,  C }
) )  =  { B ,  C }
) )
35 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( ( F `  2 )  =  ( `' E `  { C ,  A } )  ->  ( E `  ( F `  2 ) )  =  ( E `  ( `' E `  { C ,  A } ) ) )
36353ad2ant3 1019 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  2 )
)  =  ( E `
 ( `' E `  { C ,  A } ) ) )
3736eqeq1d 2469 . . . . . . . . . . . 12  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( E `  ( F `  2 ) )  =  { C ,  A }  <->  ( E `  ( `' E `  { C ,  A }
) )  =  { C ,  A }
) )
3831, 34, 373anbi123d 1299 . . . . . . . . . . 11  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } )  <->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) ) )
3928, 38ax-mp 5 . . . . . . . . . 10  |-  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } )  <->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
4027, 39sylibr 212 . . . . . . . . 9  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } ) )
41 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  0
)  =  A )
42 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  1
)  =  B )
4341, 42preq12d 4114 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
0 ) ,  ( P `  1 ) }  =  { A ,  B } )
4443eqeq2d 2481 . . . . . . . . . . 11  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  <->  ( E `  ( F `  0
) )  =  { A ,  B }
) )
45 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  2
)  =  C )
4642, 45preq12d 4114 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
1 ) ,  ( P `  2 ) }  =  { B ,  C } )
4746eqeq2d 2481 . . . . . . . . . . 11  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P ` 
2 ) }  <->  ( E `  ( F `  1
) )  =  { B ,  C }
) )
48 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  3
)  =  A )
4945, 48preq12d 4114 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
2 ) ,  ( P `  3 ) }  =  { C ,  A } )
5049eqeq2d 2481 . . . . . . . . . . 11  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  <->  ( E `  ( F `  2
) )  =  { C ,  A }
) )
5144, 47, 503anbi123d 1299 . . . . . . . . . 10  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } )  <->  ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } ) ) )
5251ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } )  <->  ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } ) ) )
5340, 52mpbird 232 . . . . . . . 8  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) )
5453ex 434 . . . . . . 7  |-  ( ( ( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  ->  (
( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) ) )
5554ex 434 . . . . . 6  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
)  ->  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) ) ) )
5615, 55mpcom 36 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) ) )
57563adant3 1016 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V )  /\  ( C  e.  V  /\  A  e.  V )
)  ->  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) ) )
588, 57mpcom 36 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) )
59 0z 10875 . . . 4  |-  0  e.  ZZ
60 1z 10894 . . . 4  |-  1  e.  ZZ
61 2z 10896 . . . 4  |-  2  e.  ZZ
62 fveq2 5866 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
6362fveq2d 5870 . . . . . 6  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
64 fveq2 5866 . . . . . . 7  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
65 oveq1 6291 . . . . . . . . 9  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
66 0p1e1 10647 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6765, 66syl6eq 2524 . . . . . . . 8  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
6867fveq2d 5870 . . . . . . 7  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
6964, 68preq12d 4114 . . . . . 6  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
7063, 69eqeq12d 2489 . . . . 5  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
71 fveq2 5866 . . . . . . 7  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
7271fveq2d 5870 . . . . . 6  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
73 fveq2 5866 . . . . . . 7  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
74 oveq1 6291 . . . . . . . . 9  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
75 1p1e2 10649 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
7674, 75syl6eq 2524 . . . . . . . 8  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
7776fveq2d 5870 . . . . . . 7  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
7873, 77preq12d 4114 . . . . . 6  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
7972, 78eqeq12d 2489 . . . . 5  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
80 fveq2 5866 . . . . . . 7  |-  ( k  =  2  ->  ( F `  k )  =  ( F ` 
2 ) )
8180fveq2d 5870 . . . . . 6  |-  ( k  =  2  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  2 )
) )
82 fveq2 5866 . . . . . . 7  |-  ( k  =  2  ->  ( P `  k )  =  ( P ` 
2 ) )
83 oveq1 6291 . . . . . . . . 9  |-  ( k  =  2  ->  (
k  +  1 )  =  ( 2  +  1 ) )
84 2p1e3 10659 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
8583, 84syl6eq 2524 . . . . . . . 8  |-  ( k  =  2  ->  (
k  +  1 )  =  3 )
8685fveq2d 5870 . . . . . . 7  |-  ( k  =  2  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
3 ) )
8782, 86preq12d 4114 . . . . . 6  |-  ( k  =  2  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
2 ) ,  ( P `  3 ) } )
8881, 87eqeq12d 2489 . . . . 5  |-  ( k  =  2  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  2
) )  =  {
( P `  2
) ,  ( P `
 3 ) } ) )
8970, 79, 88raltpg 4078 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ  /\  2  e.  ZZ )  ->  ( A. k  e.  { 0 ,  1 ,  2 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) ) )
9059, 60, 61, 89mp3an 1324 . . 3  |-  ( A. k  e.  { 0 ,  1 ,  2 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) )
9158, 90sylibr 212 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  A. k  e.  {
0 ,  1 ,  2 }  ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
9212, 13constr3lem2 24350 . . . . . 6  |-  ( # `  F )  =  3
9392oveq2i 6295 . . . . 5  |-  ( 0..^ ( # `  F
) )  =  ( 0..^ 3 )
94 fzo0to3tp 11868 . . . . 5  |-  ( 0..^ 3 )  =  {
0 ,  1 ,  2 }
9593, 94eqtri 2496 . . . 4  |-  ( 0..^ ( # `  F
) )  =  {
0 ,  1 ,  2 }
9695a1i 11 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( 0..^ (
# `  F )
)  =  { 0 ,  1 ,  2 } )
9796raleqdv 3064 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  <->  A. k  e.  { 0 ,  1 ,  2 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
9891, 97mpbird 232 1  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473    u. cun 3474   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   ran crn 5000   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    + caddc 9495   2c2 10585   3c3 10586   ZZcz 10864  ..^cfzo 11792   #chash 12373   USGrph cusg 24034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-usgra 24037
This theorem is referenced by:  constr3trl  24363
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