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 XVN0wWordX|w=N=X0..^N
4 3 ancoms N0XVwWordX|w=N=X0..^N
5 4 feq2d N0XVF:wWordX|w=NXF:X0..^NX
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 |-