Step |
Hyp |
Ref |
Expression |
1 |
|
r1funlim |
|- ( Fun R1 /\ Lim dom R1 ) |
2 |
1
|
simpri |
|- Lim dom R1 |
3 |
|
limord |
|- ( Lim dom R1 -> Ord dom R1 ) |
4 |
|
ordsson |
|- ( Ord dom R1 -> dom R1 C_ On ) |
5 |
2 3 4
|
mp2b |
|- dom R1 C_ On |
6 |
5
|
sseli |
|- ( A e. dom R1 -> A e. On ) |
7 |
5
|
sseli |
|- ( B e. dom R1 -> B e. On ) |
8 |
|
onsseleq |
|- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) ) |
9 |
6 7 8
|
syl2an |
|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) ) |
10 |
|
r1tr |
|- Tr ( R1 ` B ) |
11 |
|
r1ordg |
|- ( B e. dom R1 -> ( A e. B -> ( R1 ` A ) e. ( R1 ` B ) ) ) |
12 |
11
|
adantl |
|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( A e. B -> ( R1 ` A ) e. ( R1 ` B ) ) ) |
13 |
|
trss |
|- ( Tr ( R1 ` B ) -> ( ( R1 ` A ) e. ( R1 ` B ) -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |
14 |
10 12 13
|
mpsylsyld |
|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( A e. B -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |
15 |
|
fveq2 |
|- ( A = B -> ( R1 ` A ) = ( R1 ` B ) ) |
16 |
|
eqimss |
|- ( ( R1 ` A ) = ( R1 ` B ) -> ( R1 ` A ) C_ ( R1 ` B ) ) |
17 |
15 16
|
syl |
|- ( A = B -> ( R1 ` A ) C_ ( R1 ` B ) ) |
18 |
17
|
a1i |
|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( A = B -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |
19 |
14 18
|
jaod |
|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( ( A e. B \/ A = B ) -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |
20 |
9 19
|
sylbid |
|- ( ( A e. dom R1 /\ B e. dom R1 ) -> ( A C_ B -> ( R1 ` A ) C_ ( R1 ` B ) ) ) |