Metamath Proof Explorer

Theorem fucofn2

Description: The morphism part of the functor composition bifunctor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 29-Sep-2025)

Ref Expression
Hypotheses fucofval.c 
|- ( ph -> C e. T )
fucofval.d 
|- ( ph -> D e. U )
fucofval.e 
|- ( ph -> E e. V )
fuco1.o 
|- ( ph -> ( <. C , D >. o.F E ) = <. O , P >. )
fuco1.w 
|- ( ph -> W = ( ( D Func E ) X. ( C Func D ) ) )
Assertion fucofn2 
|- ( ph -> P Fn ( W X. W ) )

Proof

Step Hyp Ref Expression
1 fucofval.c 
 |-  ( ph -> C e. T )
2 fucofval.d 
 |-  ( ph -> D e. U )
3 fucofval.e 
 |-  ( ph -> E e. V )
4 fuco1.o 
 |-  ( ph -> ( <. C , D >. o.F E ) = <. O , P >. )
5 fuco1.w 
 |-  ( ph -> W = ( ( D Func E ) X. ( C Func D ) ) )
6 eqid 
 |-  ( u e. W , v e. W |-> [_ ( 1st ` ( 2nd ` u ) ) / f ]_ [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) ) = ( u e. W , v e. W |-> [_ ( 1st ` ( 2nd ` u ) ) / f ]_ [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) )
7 ovex 
 |-  ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) e. _V
8 ovex 
 |-  ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) e. _V
9 7 8 mpoex 
 |-  ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) e. _V
10 9 csbex 
 |-  [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) e. _V
11 10 csbex 
 |-  [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) e. _V
12 11 csbex 
 |-  [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) e. _V
13 12 csbex 
 |-  [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) e. _V
14 13 csbex 
 |-  [_ ( 1st ` ( 2nd ` u ) ) / f ]_ [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) e. _V
15 6 14 fnmpoi 
 |-  ( u e. W , v e. W |-> [_ ( 1st ` ( 2nd ` u ) ) / f ]_ [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) ) Fn ( W X. W )
16 1 2 3 4 5 fuco2 
 |-  ( ph -> P = ( u e. W , v e. W |-> [_ ( 1st ` ( 2nd ` u ) ) / f ]_ [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) ) )
17 16 fneq1d 
 |-  ( ph -> ( P Fn ( W X. W ) <-> ( u e. W , v e. W |-> [_ ( 1st ` ( 2nd ` u ) ) / f ]_ [_ ( 1st ` ( 1st ` u ) ) / k ]_ [_ ( 2nd ` ( 1st ` u ) ) / l ]_ [_ ( 1st ` ( 2nd ` v ) ) / m ]_ [_ ( 1st ` ( 1st ` v ) ) / r ]_ ( b e. ( ( 1st ` u ) ( D Nat E ) ( 1st ` v ) ) , a e. ( ( 2nd ` u ) ( C Nat D ) ( 2nd ` v ) ) |-> ( x e. ( Base ` C ) |-> ( ( b ` ( m ` x ) ) ( <. ( k ` ( f ` x ) ) , ( k ` ( m ` x ) ) >. ( comp ` E ) ( r ` ( m ` x ) ) ) ( ( ( f ` x ) l ( m ` x ) ) ` ( a ` x ) ) ) ) ) ) Fn ( W X. W ) ) )
18 15 17 mpbiri 
 |-  ( ph -> P Fn ( W X. W ) )