Metamath Proof Explorer


Theorem setsnidel

Description: The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021)

Ref Expression
Hypotheses setsidel.s
|- ( ph -> S e. V )
setsidel.b
|- ( ph -> B e. W )
setsidel.r
|- R = ( S sSet <. A , B >. )
setsnidel.c
|- ( ph -> C e. X )
setsnidel.d
|- ( ph -> D e. Y )
setsnidel.s
|- ( ph -> <. C , D >. e. S )
setsnidel.n
|- ( ph -> A =/= C )
Assertion setsnidel
|- ( ph -> <. C , D >. e. R )

Proof

Step Hyp Ref Expression
1 setsidel.s
 |-  ( ph -> S e. V )
2 setsidel.b
 |-  ( ph -> B e. W )
3 setsidel.r
 |-  R = ( S sSet <. A , B >. )
4 setsnidel.c
 |-  ( ph -> C e. X )
5 setsnidel.d
 |-  ( ph -> D e. Y )
6 setsnidel.s
 |-  ( ph -> <. C , D >. e. S )
7 setsnidel.n
 |-  ( ph -> A =/= C )
8 4 elexd
 |-  ( ph -> C e. _V )
9 7 necomd
 |-  ( ph -> C =/= A )
10 eldifsn
 |-  ( C e. ( _V \ { A } ) <-> ( C e. _V /\ C =/= A ) )
11 8 9 10 sylanbrc
 |-  ( ph -> C e. ( _V \ { A } ) )
12 opelres
 |-  ( D e. Y -> ( <. C , D >. e. ( S |` ( _V \ { A } ) ) <-> ( C e. ( _V \ { A } ) /\ <. C , D >. e. S ) ) )
13 5 12 syl
 |-  ( ph -> ( <. C , D >. e. ( S |` ( _V \ { A } ) ) <-> ( C e. ( _V \ { A } ) /\ <. C , D >. e. S ) ) )
14 11 6 13 mpbir2and
 |-  ( ph -> <. C , D >. e. ( S |` ( _V \ { A } ) ) )
15 elun1
 |-  ( <. C , D >. e. ( S |` ( _V \ { A } ) ) -> <. C , D >. e. ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
16 14 15 syl
 |-  ( ph -> <. C , D >. e. ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
17 setsval
 |-  ( ( S e. V /\ B e. W ) -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
18 1 2 17 syl2anc
 |-  ( ph -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
19 3 18 syl5eq
 |-  ( ph -> R = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
20 16 19 eleqtrrd
 |-  ( ph -> <. C , D >. e. R )