Metamath Proof Explorer

Theorem reldmprcof1

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

Ref Expression
Assertion reldmprcof1 
|- Rel dom ( 1st ` ( P -o.F F ) )

Proof

Step Hyp Ref Expression
1 relfunc 
 |-  Rel ( ( 1st ` P ) Func ( 2nd ` P ) )
2 ovex 
 |-  ( k o.func F ) e. _V
3 eqid 
 |-  ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) = ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) )
4 2 3 dmmpti 
 |-  dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) = ( ( 1st ` P ) Func ( 2nd ` P ) )
5 4 releqi 
 |-  ( Rel dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) <-> Rel ( ( 1st ` P ) Func ( 2nd ` P ) ) )
6 1 5 mpbir 
 |-  Rel dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) )
7 eqid 
 |-  ( ( 1st ` P ) Func ( 2nd ` P ) ) = ( ( 1st ` P ) Func ( 2nd ` P ) )
8 eqid 
 |-  ( ( 1st ` P ) Nat ( 2nd ` P ) ) = ( ( 1st ` P ) Nat ( 2nd ` P ) )
9 simpr 
 |-  ( ( P e. _V /\ F e. _V ) -> F e. _V )
10 simpl 
 |-  ( ( P e. _V /\ F e. _V ) -> P e. _V )
11 eqidd 
 |-  ( ( P e. _V /\ F e. _V ) -> ( 1st ` P ) = ( 1st ` P ) )
12 eqidd 
 |-  ( ( P e. _V /\ F e. _V ) -> ( 2nd ` P ) = ( 2nd ` P ) )
13 7 8 9 10 11 12 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 ) ) ) ) >. )
14 ovex 
 |-  ( ( 1st ` P ) Func ( 2nd ` P ) ) e. _V
15 14 mptex 
 |-  ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) e. _V
16 14 14 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
17 15 16 op1std 
 |-  ( ( 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 ) ) ) ) >. -> ( 1st ` ( P -o.F F ) ) = ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) )
18 13 17 syl 
 |-  ( ( P e. _V /\ F e. _V ) -> ( 1st ` ( P -o.F F ) ) = ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) )
19 18 dmeqd 
 |-  ( ( P e. _V /\ F e. _V ) -> dom ( 1st ` ( P -o.F F ) ) = dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) )
20 19 releqd 
 |-  ( ( P e. _V /\ F e. _V ) -> ( Rel dom ( 1st ` ( P -o.F F ) ) <-> Rel dom ( k e. ( ( 1st ` P ) Func ( 2nd ` P ) ) |-> ( k o.func F ) ) ) )
21 6 20 mpbiri 
 |-  ( ( P e. _V /\ F e. _V ) -> Rel dom ( 1st ` ( P -o.F F ) ) )
22 rel0 
 |-  Rel (/)
23 reldmprcof 
 |-  Rel dom -o.F
24 23 ovprc 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( P -o.F F ) = (/) )
25 24 fveq2d 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( 1st ` ( P -o.F F ) ) = ( 1st ` (/) ) )
26 1st0 
 |-  ( 1st ` (/) ) = (/)
27 25 26 eqtrdi 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( 1st ` ( P -o.F F ) ) = (/) )
28 27 dmeqd 
 |-  ( -. ( P e. _V /\ F e. _V ) -> dom ( 1st ` ( P -o.F F ) ) = dom (/) )
29 dm0 
 |-  dom (/) = (/)
30 28 29 eqtrdi 
 |-  ( -. ( P e. _V /\ F e. _V ) -> dom ( 1st ` ( P -o.F F ) ) = (/) )
31 30 releqd 
 |-  ( -. ( P e. _V /\ F e. _V ) -> ( Rel dom ( 1st ` ( P -o.F F ) ) <-> Rel (/) ) )
32 22 31 mpbiri 
 |-  ( -. ( P e. _V /\ F e. _V ) -> Rel dom ( 1st ` ( P -o.F F ) ) )
33 21 32 pm2.61i 
 |-  Rel dom ( 1st ` ( P -o.F F ) )