Theorem naryfvalelfv

Description: The value of an n-ary (endo)function on a set X is an element of X . (Contributed by AV, 14-May-2024)

	Ref	Expression
Hypothesis	naryfval.i	|- I = ( 0 ..^ N )
Assertion	naryfvalelfv	|- ( ( F e. ( N -aryF X ) /\ A : I --> X ) -> ( F ` A ) e. X )

Proof

Step	Hyp	Ref	Expression
1		naryfval.i	|- I = ( 0 ..^ N )
2	1	naryrcl	|- ( F e. ( N -aryF X ) -> ( N e. NN0 /\ X e. _V ) )
3	1	naryfvalel	|- ( ( N e. NN0 /\ X e. _V ) -> ( F e. ( N -aryF X ) <-> F : ( X ^m I ) --> X ) )
4	3	biimpd	|- ( ( N e. NN0 /\ X e. _V ) -> ( F e. ( N -aryF X ) -> F : ( X ^m I ) --> X ) )
5	2 4	mpcom	|- ( F e. ( N -aryF X ) -> F : ( X ^m I ) --> X )
6	5	adantr	|- ( ( F e. ( N -aryF X ) /\ A : I --> X ) -> F : ( X ^m I ) --> X )
7		simpr	|- ( ( N e. NN0 /\ X e. _V ) -> X e. _V )
8	1	ovexi	|- I e. _V
9	8	a1i	|- ( ( N e. NN0 /\ X e. _V ) -> I e. _V )
10	7 9	elmapd	|- ( ( N e. NN0 /\ X e. _V ) -> ( A e. ( X ^m I ) <-> A : I --> X ) )
11	10	biimpar	|- ( ( ( N e. NN0 /\ X e. _V ) /\ A : I --> X ) -> A e. ( X ^m I ) )
12	2 11	sylan	|- ( ( F e. ( N -aryF X ) /\ A : I --> X ) -> A e. ( X ^m I ) )
13	6 12	ffvelrnd	|- ( ( F e. ( N -aryF X ) /\ A : I --> X ) -> ( F ` A ) e. X )