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Metamath Proof Explorer

Theorem ttrclresv

Description: The transitive closure of R restricted to _V is the same as the transitive closure of R itself. (Contributed by Scott Fenton, 20-Oct-2024)

Ref Expression
Assertion ttrclresv 
|- t++ ( R |` _V ) = t++ R

Proof

Step Hyp Ref Expression
1 fvex 
 |-  ( f ` a ) e. _V
2 fvex 
 |-  ( f ` suc a ) e. _V
3 2 brresi 
 |-  ( ( f ` a ) ( R |` _V ) ( f ` suc a ) <-> ( ( f ` a ) e. _V /\ ( f ` a ) R ( f ` suc a ) ) )
4 1 3 mpbiran 
 |-  ( ( f ` a ) ( R |` _V ) ( f ` suc a ) <-> ( f ` a ) R ( f ` suc a ) )
5 4 ralbii 
 |-  ( A. a e. n ( f ` a ) ( R |` _V ) ( f ` suc a ) <-> A. a e. n ( f ` a ) R ( f ` suc a ) )
6 5 3anbi3i 
 |-  ( ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) ( R |` _V ) ( f ` suc a ) ) <-> ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) R ( f ` suc a ) ) )
7 6 exbii 
 |-  ( E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) ( R |` _V ) ( f ` suc a ) ) <-> E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) R ( f ` suc a ) ) )
8 7 rexbii 
 |-  ( E. n e. ( _om \ 1o ) E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) ( R |` _V ) ( f ` suc a ) ) <-> E. n e. ( _om \ 1o ) E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) R ( f ` suc a ) ) )
9 8 opabbii 
 |-  { <. x , y >. | E. n e. ( _om \ 1o ) E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) ( R |` _V ) ( f ` suc a ) ) } = { <. x , y >. | E. n e. ( _om \ 1o ) E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) R ( f ` suc a ) ) }
10 df-ttrcl 
 |-  t++ ( R |` _V ) = { <. x , y >. | E. n e. ( _om \ 1o ) E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) ( R |` _V ) ( f ` suc a ) ) }
11 df-ttrcl 
 |-  t++ R = { <. x , y >. | E. n e. ( _om \ 1o ) E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\ A. a e. n ( f ` a ) R ( f ` suc a ) ) }
12 9 10 11 3eqtr4i 
 |-  t++ ( R |` _V ) = t++ R