| Step |
Hyp |
Ref |
Expression |
| 1 |
|
f1od.1 |
|- F = ( x e. A |-> C ) |
| 2 |
|
f1od.2 |
|- ( ( ph /\ x e. A ) -> C e. W ) |
| 3 |
|
f1od.3 |
|- ( ( ph /\ y e. B ) -> D e. X ) |
| 4 |
|
f1od.4 |
|- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) ) |
| 5 |
2
|
ralrimiva |
|- ( ph -> A. x e. A C e. W ) |
| 6 |
1
|
fnmpt |
|- ( A. x e. A C e. W -> F Fn A ) |
| 7 |
5 6
|
syl |
|- ( ph -> F Fn A ) |
| 8 |
3
|
ralrimiva |
|- ( ph -> A. y e. B D e. X ) |
| 9 |
|
eqid |
|- ( y e. B |-> D ) = ( y e. B |-> D ) |
| 10 |
9
|
fnmpt |
|- ( A. y e. B D e. X -> ( y e. B |-> D ) Fn B ) |
| 11 |
8 10
|
syl |
|- ( ph -> ( y e. B |-> D ) Fn B ) |
| 12 |
4
|
opabbidv |
|- ( ph -> { <. y , x >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( y e. B /\ x = D ) } ) |
| 13 |
|
df-mpt |
|- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) } |
| 14 |
1 13
|
eqtri |
|- F = { <. x , y >. | ( x e. A /\ y = C ) } |
| 15 |
14
|
cnveqi |
|- `' F = `' { <. x , y >. | ( x e. A /\ y = C ) } |
| 16 |
|
cnvopab |
|- `' { <. x , y >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( x e. A /\ y = C ) } |
| 17 |
15 16
|
eqtri |
|- `' F = { <. y , x >. | ( x e. A /\ y = C ) } |
| 18 |
|
df-mpt |
|- ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) } |
| 19 |
12 17 18
|
3eqtr4g |
|- ( ph -> `' F = ( y e. B |-> D ) ) |
| 20 |
19
|
fneq1d |
|- ( ph -> ( `' F Fn B <-> ( y e. B |-> D ) Fn B ) ) |
| 21 |
11 20
|
mpbird |
|- ( ph -> `' F Fn B ) |
| 22 |
|
dff1o4 |
|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) ) |
| 23 |
7 21 22
|
sylanbrc |
|- ( ph -> F : A -1-1-onto-> B ) |
| 24 |
23 19
|
jca |
|- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) ) |