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Metamath Proof Explorer

Theorem fsets

Description: The structure replacement function is a function. (Contributed by SO, 12-Jul-2018)

Ref Expression
Assertion fsets 
|- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( F sSet <. X , Y >. ) : A --> B )

Proof

Step Hyp Ref Expression
1 difss 
 |-  ( A \ { X } ) C_ A
2 fssres 
 |-  ( ( F : A --> B /\ ( A \ { X } ) C_ A ) -> ( F |` ( A \ { X } ) ) : ( A \ { X } ) --> B )
3 1 2 mpan2 
 |-  ( F : A --> B -> ( F |` ( A \ { X } ) ) : ( A \ { X } ) --> B )
4 ffn 
 |-  ( F : A --> B -> F Fn A )
5 fnresdm 
 |-  ( F Fn A -> ( F |` A ) = F )
6 4 5 syl 
 |-  ( F : A --> B -> ( F |` A ) = F )
7 6 reseq1d 
 |-  ( F : A --> B -> ( ( F |` A ) |` ( _V \ { X } ) ) = ( F |` ( _V \ { X } ) ) )
8 resres 
 |-  ( ( F |` A ) |` ( _V \ { X } ) ) = ( F |` ( A i^i ( _V \ { X } ) ) )
9 invdif 
 |-  ( A i^i ( _V \ { X } ) ) = ( A \ { X } )
10 9 reseq2i 
 |-  ( F |` ( A i^i ( _V \ { X } ) ) ) = ( F |` ( A \ { X } ) )
11 8 10 eqtri 
 |-  ( ( F |` A ) |` ( _V \ { X } ) ) = ( F |` ( A \ { X } ) )
12 7 11 eqtr3di 
 |-  ( F : A --> B -> ( F |` ( _V \ { X } ) ) = ( F |` ( A \ { X } ) ) )
13 12 feq1d 
 |-  ( F : A --> B -> ( ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B <-> ( F |` ( A \ { X } ) ) : ( A \ { X } ) --> B ) )
14 3 13 mpbird 
 |-  ( F : A --> B -> ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B )
15 14 adantl 
 |-  ( ( F e. V /\ F : A --> B ) -> ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B )
16 fsnunf2 
 |-  ( ( ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B /\ X e. A /\ Y e. B ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B )
17 15 16 syl3an1 
 |-  ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B )
18 simp1l 
 |-  ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> F e. V )
19 simp3 
 |-  ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> Y e. B )
20 setsval 
 |-  ( ( F e. V /\ Y e. B ) -> ( F sSet <. X , Y >. ) = ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) )
21 20 feq1d 
 |-  ( ( F e. V /\ Y e. B ) -> ( ( F sSet <. X , Y >. ) : A --> B <-> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B ) )
22 18 19 21 syl2anc 
 |-  ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( ( F sSet <. X , Y >. ) : A --> B <-> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B ) )
23 17 22 mpbird 
 |-  ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( F sSet <. X , Y >. ) : A --> B )