Step |
Hyp |
Ref |
Expression |
1 |
|
dmresv |
|- dom ( A |` _V ) = dom A |
2 |
|
resss |
|- ( A |` _V ) C_ A |
3 |
|
ctex |
|- ( A ~<_ _om -> A e. _V ) |
4 |
|
ssexg |
|- ( ( ( A |` _V ) C_ A /\ A e. _V ) -> ( A |` _V ) e. _V ) |
5 |
2 3 4
|
sylancr |
|- ( A ~<_ _om -> ( A |` _V ) e. _V ) |
6 |
|
fvex |
|- ( 1st ` x ) e. _V |
7 |
|
eqid |
|- ( x e. ( A |` _V ) |-> ( 1st ` x ) ) = ( x e. ( A |` _V ) |-> ( 1st ` x ) ) |
8 |
6 7
|
fnmpti |
|- ( x e. ( A |` _V ) |-> ( 1st ` x ) ) Fn ( A |` _V ) |
9 |
|
dffn4 |
|- ( ( x e. ( A |` _V ) |-> ( 1st ` x ) ) Fn ( A |` _V ) <-> ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> ran ( x e. ( A |` _V ) |-> ( 1st ` x ) ) ) |
10 |
8 9
|
mpbi |
|- ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> ran ( x e. ( A |` _V ) |-> ( 1st ` x ) ) |
11 |
|
relres |
|- Rel ( A |` _V ) |
12 |
|
reldm |
|- ( Rel ( A |` _V ) -> dom ( A |` _V ) = ran ( x e. ( A |` _V ) |-> ( 1st ` x ) ) ) |
13 |
|
foeq3 |
|- ( dom ( A |` _V ) = ran ( x e. ( A |` _V ) |-> ( 1st ` x ) ) -> ( ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> dom ( A |` _V ) <-> ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> ran ( x e. ( A |` _V ) |-> ( 1st ` x ) ) ) ) |
14 |
11 12 13
|
mp2b |
|- ( ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> dom ( A |` _V ) <-> ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> ran ( x e. ( A |` _V ) |-> ( 1st ` x ) ) ) |
15 |
10 14
|
mpbir |
|- ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> dom ( A |` _V ) |
16 |
|
fodomg |
|- ( ( A |` _V ) e. _V -> ( ( x e. ( A |` _V ) |-> ( 1st ` x ) ) : ( A |` _V ) -onto-> dom ( A |` _V ) -> dom ( A |` _V ) ~<_ ( A |` _V ) ) ) |
17 |
5 15 16
|
mpisyl |
|- ( A ~<_ _om -> dom ( A |` _V ) ~<_ ( A |` _V ) ) |
18 |
|
ssdomg |
|- ( A e. _V -> ( ( A |` _V ) C_ A -> ( A |` _V ) ~<_ A ) ) |
19 |
3 2 18
|
mpisyl |
|- ( A ~<_ _om -> ( A |` _V ) ~<_ A ) |
20 |
|
domtr |
|- ( ( ( A |` _V ) ~<_ A /\ A ~<_ _om ) -> ( A |` _V ) ~<_ _om ) |
21 |
19 20
|
mpancom |
|- ( A ~<_ _om -> ( A |` _V ) ~<_ _om ) |
22 |
|
domtr |
|- ( ( dom ( A |` _V ) ~<_ ( A |` _V ) /\ ( A |` _V ) ~<_ _om ) -> dom ( A |` _V ) ~<_ _om ) |
23 |
17 21 22
|
syl2anc |
|- ( A ~<_ _om -> dom ( A |` _V ) ~<_ _om ) |
24 |
1 23
|
eqbrtrrid |
|- ( A ~<_ _om -> dom A ~<_ _om ) |