Metamath Proof Explorer


Theorem naryfvalelwrdf

Description: An n-ary (endo)function on a set X expressed as a function over the set of words on X of length n . (Contributed by AV, 4-Jun-2024)

Ref Expression
Assertion naryfvalelwrdf Could not format assertion : No typesetting found for |- ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : { w e. Word X | ( # ` w ) = N } --> X ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 eqid 0 ..^ N = 0 ..^ N
2 1 naryfvalel Could not format ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : ( X ^m ( 0 ..^ N ) ) --> X ) ) : No typesetting found for |- ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : ( X ^m ( 0 ..^ N ) ) --> X ) ) with typecode |-
3 wrdnval X V N 0 w Word X | w = N = X 0 ..^ N
4 3 ancoms N 0 X V w Word X | w = N = X 0 ..^ N
5 4 feq2d N 0 X V F : w Word X | w = N X F : X 0 ..^ N X
6 2 5 bitr4d Could not format ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : { w e. Word X | ( # ` w ) = N } --> X ) ) : No typesetting found for |- ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : { w e. Word X | ( # ` w ) = N } --> X ) ) with typecode |-