Metamath Proof Explorer

Theorem reldmprcof2

Description: The domain of the morphism part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025)

Ref Expression
Assertion reldmprcof2 
|- Rel dom ( 2nd ` ( P -o.F F ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) = ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) )
2 1 reldmmpo 
 |-  Rel dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) )
3 eqid 
 |-  ( ( 1st ` P ) Func ( 2nd ` P ) ) = ( ( 1st ` P ) Func ( 2nd ` P ) )
4 eqid 
 |-  ( ( 1st ` P ) Nat ( 2nd ` P ) ) = ( ( 1st ` P ) Nat ( 2nd ` P ) )
5 simpr 
 |-  ( ( P e. _V /\ F e. _V ) -> F e. _V )
6 simpl 
 |-  ( ( P e. _V /\ F e. _V ) -> P e. _V )
7 eqidd 
 |-  ( ( P e. _V /\ F e. _V ) -> ( 1st ` P ) = ( 1st ` P ) )
8 eqidd 
 |-  ( ( P e. _V /\ F e. _V ) -> ( 2nd ` P ) = ( 2nd ` P ) )
9 3 4 5 6 7 8 prcofvalg 
 |-  ( ( P e. _V /\ F e. _V ) -> ( P -o.F F ) = <. ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) , ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) >. )
10 ovex 
 |-  ( ( 1st ` P ) Func ( 2nd ` P ) ) e. _V
11 10 mptex 
 |-  ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) e. _V
12 10 10 mpoex 
 |-  ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) e. _V
13 11 12 op2ndd 
 |-  ( ( P -o.F F ) = <. ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) , ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) >. -> ( 2nd ` ( P -o.F F ) ) = ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) )
14 9 13 syl 
 |-  ( ( P e. _V /\ F e. _V ) -> ( 2nd ` ( P -o.F F ) ) = ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) )
15 14 dmeqd 
 |-  ( ( P e. _V /\ F e. _V ) -> dom ( 2nd ` ( P -o.F F ) ) = dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) )
16 15 releqd 
 |-  ( ( P e. _V /\ F e. _V ) -> ( Rel dom ( 2nd ` ( P -o.F F ) ) <-> Rel dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) , l e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( a e. ( k ( ( 1st ` P ) Nat ( 2nd ` P ) ) l ) |-> ( a o. ( 1st ` F ) ) ) ) ) )
17 2 16 mpbiri 
 |-  ( ( P e. _V /\ F e. _V ) -> Rel dom ( 2nd ` ( P -o.F F ) ) )
18 rel0 
 |-  Rel (/)
19 reldmprcof 
 |-  Rel dom -o.F
20 19 ovprc 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( P -o.F F ) = (/) )
21 20 fveq2d 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( 2nd ` ( P -o.F F ) ) = ( 2nd ` (/) ) )
22 2nd0 
 |-  ( 2nd ` (/) ) = (/)
23 21 22 eqtrdi 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( 2nd ` ( P -o.F F ) ) = (/) )
24 23 dmeqd 
 |-  ( -. ( P e. _V /\ F e. _V ) -> dom ( 2nd ` ( P -o.F F ) ) = dom (/) )
25 dm0 
 |-  dom (/) = (/)
26 24 25 eqtrdi 
 |-  ( -. ( P e. _V /\ F e. _V ) -> dom ( 2nd ` ( P -o.F F ) ) = (/) )
27 26 releqd 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( Rel dom ( 2nd ` ( P -o.F F ) ) <-> Rel (/) ) )
28 18 27 mpbiri 
 |-  ( -. ( P e. _V /\ F e. _V ) -> Rel dom ( 2nd ` ( P -o.F F ) ) )
29 17 28 pm2.61i 
 |-  Rel dom ( 2nd ` ( P -o.F F ) )