Theorem setsn0fun

Description: The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021) (Revised by AV, 16-Nov-2021)

	Ref	Expression
Hypotheses	setsn0fun.s	|- ( ph -> S Struct X )
	setsn0fun.i	|- ( ph -> I e. U )
	setsn0fun.e	|- ( ph -> E e. W )
Assertion	setsn0fun	|- ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )

Proof

Step	Hyp	Ref	Expression
1		setsn0fun.s	|- ( ph -> S Struct X )
2		setsn0fun.i	|- ( ph -> I e. U )
3		setsn0fun.e	|- ( ph -> E e. W )
4		structn0fun	|- ( S Struct X -> Fun ( S \ { (/) } ) )
5		structex	|- ( S Struct X -> S e. _V )
6		setsfun0	|- ( ( ( S e. _V /\ Fun ( S \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )
7	5 6	sylanl1	|- ( ( ( S Struct X /\ Fun ( S \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )
8	7	expcom	|- ( ( I e. U /\ E e. W ) -> ( ( S Struct X /\ Fun ( S \ { (/) } ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
9	2 3 8	syl2anc	|- ( ph -> ( ( S Struct X /\ Fun ( S \ { (/) } ) ) -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
10	9	com12	|- ( ( S Struct X /\ Fun ( S \ { (/) } ) ) -> ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
11	4 10	mpdan	|- ( S Struct X -> ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) ) )
12	1 11	mpcom	|- ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )