Metamath Proof Explorer

Theorem stgrfv

Description: The star graph S_N. (Contributed by AV, 10-Sep-2025)

Ref Expression
Assertion stgrfv 
|- ( N e. NN0 -> ( StarGr ` N ) = { <. ( Base ` ndx ) , ( 0 ... N ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. } )

Proof

Step Hyp Ref Expression
1 df-stgr 
 |-  StarGr = ( n e. NN0 |-> { <. ( Base ` ndx ) , ( 0 ... n ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } ) >. } )
2 1 a1i 
 |-  ( N e. NN0 -> StarGr = ( n e. NN0 |-> { <. ( Base ` ndx ) , ( 0 ... n ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } ) >. } ) )
3 oveq2 
 |-  ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
4 3 opeq2d 
 |-  ( n = N -> <. ( Base ` ndx ) , ( 0 ... n ) >. = <. ( Base ` ndx ) , ( 0 ... N ) >. )
5 3 pweqd 
 |-  ( n = N -> ~P ( 0 ... n ) = ~P ( 0 ... N ) )
6 oveq2 
 |-  ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
7 6 rexeqdv 
 |-  ( n = N -> ( E. x e. ( 1 ... n ) e = { 0 , x } <-> E. x e. ( 1 ... N ) e = { 0 , x } ) )
8 5 7 rabeqbidv 
 |-  ( n = N -> { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } = { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } )
9 8 reseq2d 
 |-  ( n = N -> ( _I |` { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } ) = ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) )
10 9 opeq2d 
 |-  ( n = N -> <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } ) >. = <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. )
11 4 10 preq12d 
 |-  ( n = N -> { <. ( Base ` ndx ) , ( 0 ... n ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } ) >. } = { <. ( Base ` ndx ) , ( 0 ... N ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. } )
12 11 adantl 
 |-  ( ( N e. NN0 /\ n = N ) -> { <. ( Base ` ndx ) , ( 0 ... n ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... n ) | E. x e. ( 1 ... n ) e = { 0 , x } } ) >. } = { <. ( Base ` ndx ) , ( 0 ... N ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. } )
13 id 
 |-  ( N e. NN0 -> N e. NN0 )
14 prex 
 |-  { <. ( Base ` ndx ) , ( 0 ... N ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. } e. _V
15 14 a1i 
 |-  ( N e. NN0 -> { <. ( Base ` ndx ) , ( 0 ... N ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. } e. _V )
16 2 12 13 15 fvmptd 
 |-  ( N e. NN0 -> ( StarGr ` N ) = { <. ( Base ` ndx ) , ( 0 ... N ) >. , <. ( .ef ` ndx ) , ( _I |` { e e. ~P ( 0 ... N ) | E. x e. ( 1 ... N ) e = { 0 , x } } ) >. } )