| Step |
Hyp |
Ref |
Expression |
| 1 |
|
initopropd.1 |
|- ( ph -> ( Homf ` C ) = ( Homf ` D ) ) |
| 2 |
|
initopropd.2 |
|- ( ph -> ( comf ` C ) = ( comf ` D ) ) |
| 3 |
|
initopropdlem.1 |
|- ( ph -> -. C e. _V ) |
| 4 |
|
termofn |
|- TermO Fn Cat |
| 5 |
|
ssv |
|- Cat C_ _V |
| 6 |
|
simpr |
|- ( ( ph /\ D e. Cat ) -> D e. Cat ) |
| 7 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
| 8 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
| 9 |
6 7 8
|
termoval |
|- ( ( ph /\ D e. Cat ) -> ( TermO ` D ) = { a e. ( Base ` D ) | A. b e. ( Base ` D ) E! h h e. ( b ( Hom ` D ) a ) } ) |
| 10 |
|
fvprc |
|- ( -. C e. _V -> ( Homf ` C ) = (/) ) |
| 11 |
3 10
|
syl |
|- ( ph -> ( Homf ` C ) = (/) ) |
| 12 |
1 11
|
eqtr3d |
|- ( ph -> ( Homf ` D ) = (/) ) |
| 13 |
|
homf0 |
|- ( ( Base ` D ) = (/) <-> ( Homf ` D ) = (/) ) |
| 14 |
12 13
|
sylibr |
|- ( ph -> ( Base ` D ) = (/) ) |
| 15 |
14
|
rabeqdv |
|- ( ph -> { a e. ( Base ` D ) | A. b e. ( Base ` D ) E! h h e. ( b ( Hom ` D ) a ) } = { a e. (/) | A. b e. ( Base ` D ) E! h h e. ( b ( Hom ` D ) a ) } ) |
| 16 |
|
rab0 |
|- { a e. (/) | A. b e. ( Base ` D ) E! h h e. ( b ( Hom ` D ) a ) } = (/) |
| 17 |
15 16
|
eqtrdi |
|- ( ph -> { a e. ( Base ` D ) | A. b e. ( Base ` D ) E! h h e. ( b ( Hom ` D ) a ) } = (/) ) |
| 18 |
17
|
adantr |
|- ( ( ph /\ D e. Cat ) -> { a e. ( Base ` D ) | A. b e. ( Base ` D ) E! h h e. ( b ( Hom ` D ) a ) } = (/) ) |
| 19 |
9 18
|
eqtrd |
|- ( ( ph /\ D e. Cat ) -> ( TermO ` D ) = (/) ) |
| 20 |
4 3 5 19
|
initopropdlemlem |
|- ( ph -> ( TermO ` C ) = ( TermO ` D ) ) |