| Step |
Hyp |
Ref |
Expression |
| 1 |
|
idfu2nda.i |
|- I = ( idFunc ` C ) |
| 2 |
|
idfu2nda.d |
|- ( ph -> I e. ( D Func E ) ) |
| 3 |
|
idfu2nda.b |
|- ( ph -> B = ( Base ` D ) ) |
| 4 |
|
idfu2nda.x |
|- ( ph -> X e. B ) |
| 5 |
|
idfu2nda.y |
|- ( ph -> Y e. B ) |
| 6 |
|
idfu2nda.h |
|- ( ph -> H = ( X ( Hom ` D ) Y ) ) |
| 7 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
| 8 |
1 2
|
eqeltrrid |
|- ( ph -> ( idFunc ` C ) e. ( D Func E ) ) |
| 9 |
|
idfurcl |
|- ( ( idFunc ` C ) e. ( D Func E ) -> C e. Cat ) |
| 10 |
8 9
|
syl |
|- ( ph -> C e. Cat ) |
| 11 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
| 12 |
1 2 3
|
idfu1stalem |
|- ( ph -> B = ( Base ` C ) ) |
| 13 |
4 12
|
eleqtrd |
|- ( ph -> X e. ( Base ` C ) ) |
| 14 |
5 12
|
eleqtrd |
|- ( ph -> Y e. ( Base ` C ) ) |
| 15 |
1 7 10 11 13 14
|
idfu2nd |
|- ( ph -> ( X ( 2nd ` I ) Y ) = ( _I |` ( X ( Hom ` C ) Y ) ) ) |
| 16 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
| 17 |
1
|
idfucl |
|- ( C e. Cat -> I e. ( C Func C ) ) |
| 18 |
10 17
|
syl |
|- ( ph -> I e. ( C Func C ) ) |
| 19 |
18
|
func1st2nd |
|- ( ph -> ( 1st ` I ) ( C Func C ) ( 2nd ` I ) ) |
| 20 |
2
|
func1st2nd |
|- ( ph -> ( 1st ` I ) ( D Func E ) ( 2nd ` I ) ) |
| 21 |
19 20
|
funchomf |
|- ( ph -> ( Homf ` C ) = ( Homf ` D ) ) |
| 22 |
7 11 16 21 13 14
|
homfeqval |
|- ( ph -> ( X ( Hom ` C ) Y ) = ( X ( Hom ` D ) Y ) ) |
| 23 |
6 22
|
eqtr4d |
|- ( ph -> H = ( X ( Hom ` C ) Y ) ) |
| 24 |
23
|
reseq2d |
|- ( ph -> ( _I |` H ) = ( _I |` ( X ( Hom ` C ) Y ) ) ) |
| 25 |
15 24
|
eqtr4d |
|- ( ph -> ( X ( 2nd ` I ) Y ) = ( _I |` H ) ) |