Theorem naryfvalel

Description: An n-ary (endo)function on a set X . (Contributed by AV, 14-May-2024)

	Ref	Expression
Hypothesis	naryfval.i	$$⊢I=\left(0..^N\right)$$
Assertion	naryfvalel	Could not format assertion : No typesetting found for |- ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : ( X ^m I ) --> X ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		naryfval.i	$$⊢I=\left(0..^N\right)$$
2	1	naryfval	Could not format ( N e. NN0 -> ( N -aryF X ) = ( X ^m ( X ^m I ) ) ) : No typesetting found for |- ( N e. NN0 -> ( N -aryF X ) = ( X ^m ( X ^m I ) ) ) with typecode |-
3	2	eleq2d	Could not format ( N e. NN0 -> ( F e. ( N -aryF X ) <-> F e. ( X ^m ( X ^m I ) ) ) ) : No typesetting found for |- ( N e. NN0 -> ( F e. ( N -aryF X ) <-> F e. ( X ^m ( X ^m I ) ) ) ) with typecode |-
4		ovex	$$⊢X^I\in \mathrm{V}$$
5		elmapg	$$⊢\left(X\in V\wedge X^I\in \mathrm{V}\right)\to \left(F\in \left(X^{\left(X^I\right)}\right)\leftrightarrow F:X^I⟶X\right)$$
6	4 5	mpan2	$$⊢X\in V\to \left(F\in \left(X^{\left(X^I\right)}\right)\leftrightarrow F:X^I⟶X\right)$$
7	3 6	sylan9bb	Could not format ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : ( X ^m I ) --> X ) ) : No typesetting found for |- ( ( N e. NN0 /\ X e. V ) -> ( F e. ( N -aryF X ) <-> F : ( X ^m I ) --> X ) ) with typecode |-