Step |
Hyp |
Ref |
Expression |
1 |
|
reldom |
|- Rel ~<_ |
2 |
1
|
brrelex1i |
|- ( A ~<_ B -> A e. _V ) |
3 |
|
xpcomeng |
|- ( ( A e. _V /\ C e. V ) -> ( A X. C ) ~~ ( C X. A ) ) |
4 |
3
|
ancoms |
|- ( ( C e. V /\ A e. _V ) -> ( A X. C ) ~~ ( C X. A ) ) |
5 |
2 4
|
sylan2 |
|- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~~ ( C X. A ) ) |
6 |
|
xpdom2g |
|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) ) |
7 |
1
|
brrelex2i |
|- ( A ~<_ B -> B e. _V ) |
8 |
|
xpcomeng |
|- ( ( C e. V /\ B e. _V ) -> ( C X. B ) ~~ ( B X. C ) ) |
9 |
7 8
|
sylan2 |
|- ( ( C e. V /\ A ~<_ B ) -> ( C X. B ) ~~ ( B X. C ) ) |
10 |
|
domentr |
|- ( ( ( C X. A ) ~<_ ( C X. B ) /\ ( C X. B ) ~~ ( B X. C ) ) -> ( C X. A ) ~<_ ( B X. C ) ) |
11 |
6 9 10
|
syl2anc |
|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( B X. C ) ) |
12 |
|
endomtr |
|- ( ( ( A X. C ) ~~ ( C X. A ) /\ ( C X. A ) ~<_ ( B X. C ) ) -> ( A X. C ) ~<_ ( B X. C ) ) |
13 |
5 11 12
|
syl2anc |
|- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) ) |