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Metamath Proof Explorer

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 >. ) \ { (/) } ) )