Metamath Proof Explorer

Theorem ranfn

Description: Ran is a function on ( (V X. V ) X. _V ) . (Contributed by Zhi Wang, 4-Nov-2025)

Ref Expression
Assertion ranfn Could not format assertion : No typesetting found for |- Ran Fn ( ( _V X. _V ) X. _V ) with typecode |-

Proof

Step Hyp Ref Expression
1 df-ran Could not format Ran = ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) : No typesetting found for |- Ran = ( p e. ( _V X. _V ) , e e. _V |-> [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) ) with typecode |-
2 ovex ⊢ c Func d ∈ V
3 ovex ⊢ c Func e ∈ V
4 2 3 mpoex Could not format ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) e. _V : No typesetting found for |- ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) e. _V with typecode |-
5 4 csbex Could not format [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) e. _V : No typesetting found for |- [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) e. _V with typecode |-
6 5 csbex Could not format [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) e. _V : No typesetting found for |- [_ ( 1st ` p ) / c ]_ [_ ( 2nd ` p ) / d ]_ ( f e. ( c Func d ) , x e. ( c Func e ) |-> ( ( oppFunc ` ( <. d , e >. -o.F f ) ) ( ( oppCat ` ( d FuncCat e ) ) UP ( oppCat ` ( c FuncCat e ) ) ) x ) ) e. _V with typecode |-
7 1 6 fnmpoi Could not format Ran Fn ( ( _V X. _V ) X. _V ) : No typesetting found for |- Ran Fn ( ( _V X. _V ) X. _V ) with typecode |-