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 ) |