Step |
Hyp |
Ref |
Expression |
1 |
|
df-ima |
|- ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ran ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) |
2 |
|
simpr |
|- ( ( F Fn X /\ A C_ X ) -> A C_ X ) |
3 |
|
resmpo |
|- ( ( A C_ X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) |
4 |
2 3
|
sylancom |
|- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) |
5 |
4
|
rneqd |
|- ( ( F Fn X /\ A C_ X ) -> ran ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) |` ( A X. A ) ) = ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) |
6 |
1 5
|
syl5eq |
|- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) |
7 |
|
vex |
|- x e. _V |
8 |
|
vex |
|- y e. _V |
9 |
7 8
|
op1std |
|- ( p = <. x , y >. -> ( 1st ` p ) = x ) |
10 |
9
|
fveq2d |
|- ( p = <. x , y >. -> ( F ` ( 1st ` p ) ) = ( F ` x ) ) |
11 |
7 8
|
op2ndd |
|- ( p = <. x , y >. -> ( 2nd ` p ) = y ) |
12 |
11
|
fveq2d |
|- ( p = <. x , y >. -> ( F ` ( 2nd ` p ) ) = ( F ` y ) ) |
13 |
10 12
|
opeq12d |
|- ( p = <. x , y >. -> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. = <. ( F ` x ) , ( F ` y ) >. ) |
14 |
13
|
mpompt |
|- ( p e. ( A X. A ) |-> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) |
15 |
14
|
eqcomi |
|- ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ( p e. ( A X. A ) |-> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. ) |
16 |
15
|
rneqi |
|- ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ran ( p e. ( A X. A ) |-> <. ( F ` ( 1st ` p ) ) , ( F ` ( 2nd ` p ) ) >. ) |
17 |
|
fvexd |
|- ( ( T. /\ p e. ( A X. A ) ) -> ( F ` ( 1st ` p ) ) e. _V ) |
18 |
|
fvexd |
|- ( ( T. /\ p e. ( A X. A ) ) -> ( F ` ( 2nd ` p ) ) e. _V ) |
19 |
16 17 18
|
fliftrel |
|- ( T. -> ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) C_ ( _V X. _V ) ) |
20 |
19
|
mptru |
|- ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) C_ ( _V X. _V ) |
21 |
20
|
sseli |
|- ( p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) -> p e. ( _V X. _V ) ) |
22 |
21
|
adantl |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) -> p e. ( _V X. _V ) ) |
23 |
|
xpss |
|- ( ( F " A ) X. ( F " A ) ) C_ ( _V X. _V ) |
24 |
23
|
sseli |
|- ( p e. ( ( F " A ) X. ( F " A ) ) -> p e. ( _V X. _V ) ) |
25 |
24
|
adantl |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( ( F " A ) X. ( F " A ) ) ) -> p e. ( _V X. _V ) ) |
26 |
|
eqid |
|- ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) |
27 |
|
opex |
|- <. ( F ` x ) , ( F ` y ) >. e. _V |
28 |
26 27
|
elrnmpo |
|- ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> E. x e. A E. y e. A <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. ) |
29 |
|
eqcom |
|- ( <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. <-> <. ( F ` x ) , ( F ` y ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. ) |
30 |
|
fvex |
|- ( 1st ` p ) e. _V |
31 |
|
fvex |
|- ( 2nd ` p ) e. _V |
32 |
30 31
|
opth2 |
|- ( <. ( F ` x ) , ( F ` y ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. <-> ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) ) |
33 |
29 32
|
bitri |
|- ( <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. <-> ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) ) |
34 |
33
|
2rexbii |
|- ( E. x e. A E. y e. A <. ( 1st ` p ) , ( 2nd ` p ) >. = <. ( F ` x ) , ( F ` y ) >. <-> E. x e. A E. y e. A ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) ) |
35 |
|
reeanv |
|- ( E. x e. A E. y e. A ( ( F ` x ) = ( 1st ` p ) /\ ( F ` y ) = ( 2nd ` p ) ) <-> ( E. x e. A ( F ` x ) = ( 1st ` p ) /\ E. y e. A ( F ` y ) = ( 2nd ` p ) ) ) |
36 |
28 34 35
|
3bitri |
|- ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> ( E. x e. A ( F ` x ) = ( 1st ` p ) /\ E. y e. A ( F ` y ) = ( 2nd ` p ) ) ) |
37 |
|
fvelimab |
|- ( ( F Fn X /\ A C_ X ) -> ( ( 1st ` p ) e. ( F " A ) <-> E. x e. A ( F ` x ) = ( 1st ` p ) ) ) |
38 |
|
fvelimab |
|- ( ( F Fn X /\ A C_ X ) -> ( ( 2nd ` p ) e. ( F " A ) <-> E. y e. A ( F ` y ) = ( 2nd ` p ) ) ) |
39 |
37 38
|
anbi12d |
|- ( ( F Fn X /\ A C_ X ) -> ( ( ( 1st ` p ) e. ( F " A ) /\ ( 2nd ` p ) e. ( F " A ) ) <-> ( E. x e. A ( F ` x ) = ( 1st ` p ) /\ E. y e. A ( F ` y ) = ( 2nd ` p ) ) ) ) |
40 |
36 39
|
bitr4id |
|- ( ( F Fn X /\ A C_ X ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> ( ( 1st ` p ) e. ( F " A ) /\ ( 2nd ` p ) e. ( F " A ) ) ) ) |
41 |
|
opelxp |
|- ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) <-> ( ( 1st ` p ) e. ( F " A ) /\ ( 2nd ` p ) e. ( F " A ) ) ) |
42 |
40 41
|
bitr4di |
|- ( ( F Fn X /\ A C_ X ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) ) ) |
43 |
42
|
adantr |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) ) ) |
44 |
|
1st2nd2 |
|- ( p e. ( _V X. _V ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. ) |
45 |
44
|
adantl |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. ) |
46 |
45
|
eleq1d |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) ) ) |
47 |
45
|
eleq1d |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( p e. ( ( F " A ) X. ( F " A ) ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. ( ( F " A ) X. ( F " A ) ) ) ) |
48 |
43 46 47
|
3bitr4d |
|- ( ( ( F Fn X /\ A C_ X ) /\ p e. ( _V X. _V ) ) -> ( p e. ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) <-> p e. ( ( F " A ) X. ( F " A ) ) ) ) |
49 |
22 25 48
|
eqrdav |
|- ( ( F Fn X /\ A C_ X ) -> ran ( x e. A , y e. A |-> <. ( F ` x ) , ( F ` y ) >. ) = ( ( F " A ) X. ( F " A ) ) ) |
50 |
6 49
|
eqtrd |
|- ( ( F Fn X /\ A C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( A X. A ) ) = ( ( F " A ) X. ( F " A ) ) ) |