| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fcores.f |
|- ( ph -> F : A --> B ) |
| 2 |
|
fcores.e |
|- E = ( ran F i^i C ) |
| 3 |
|
fcores.p |
|- P = ( `' F " C ) |
| 4 |
|
fcores.x |
|- X = ( F |` P ) |
| 5 |
|
fcores.g |
|- ( ph -> G : C --> D ) |
| 6 |
|
fcores.y |
|- Y = ( G |` E ) |
| 7 |
|
f1cof1blem.s |
|- ( ph -> ran F = C ) |
| 8 |
7
|
eqcomd |
|- ( ph -> C = ran F ) |
| 9 |
8
|
imaeq2d |
|- ( ph -> ( `' F " C ) = ( `' F " ran F ) ) |
| 10 |
3 9
|
eqtrid |
|- ( ph -> P = ( `' F " ran F ) ) |
| 11 |
|
cnvimarndm |
|- ( `' F " ran F ) = dom F |
| 12 |
1
|
fdmd |
|- ( ph -> dom F = A ) |
| 13 |
11 12
|
eqtrid |
|- ( ph -> ( `' F " ran F ) = A ) |
| 14 |
10 13
|
eqtrd |
|- ( ph -> P = A ) |
| 15 |
|
simpr |
|- ( ( ph /\ ran F = C ) -> ran F = C ) |
| 16 |
15
|
ineq1d |
|- ( ( ph /\ ran F = C ) -> ( ran F i^i C ) = ( C i^i C ) ) |
| 17 |
|
inidm |
|- ( C i^i C ) = C |
| 18 |
16 17
|
eqtrdi |
|- ( ( ph /\ ran F = C ) -> ( ran F i^i C ) = C ) |
| 19 |
7 18
|
mpdan |
|- ( ph -> ( ran F i^i C ) = C ) |
| 20 |
2 19
|
eqtrid |
|- ( ph -> E = C ) |
| 21 |
14 20
|
jca |
|- ( ph -> ( P = A /\ E = C ) ) |
| 22 |
10 11
|
eqtrdi |
|- ( ph -> P = dom F ) |
| 23 |
22
|
reseq2d |
|- ( ph -> ( F |` P ) = ( F |` dom F ) ) |
| 24 |
4 23
|
eqtrid |
|- ( ph -> X = ( F |` dom F ) ) |
| 25 |
1
|
freld |
|- ( ph -> Rel F ) |
| 26 |
|
resdm |
|- ( Rel F -> ( F |` dom F ) = F ) |
| 27 |
25 26
|
syl |
|- ( ph -> ( F |` dom F ) = F ) |
| 28 |
24 27
|
eqtrd |
|- ( ph -> X = F ) |
| 29 |
5
|
fdmd |
|- ( ph -> dom G = C ) |
| 30 |
20 29
|
eqtr4d |
|- ( ph -> E = dom G ) |
| 31 |
30
|
reseq2d |
|- ( ph -> ( G |` E ) = ( G |` dom G ) ) |
| 32 |
6 31
|
eqtrid |
|- ( ph -> Y = ( G |` dom G ) ) |
| 33 |
5
|
freld |
|- ( ph -> Rel G ) |
| 34 |
|
resdm |
|- ( Rel G -> ( G |` dom G ) = G ) |
| 35 |
33 34
|
syl |
|- ( ph -> ( G |` dom G ) = G ) |
| 36 |
32 35
|
eqtrd |
|- ( ph -> Y = G ) |
| 37 |
21 28 36
|
jca32 |
|- ( ph -> ( ( P = A /\ E = C ) /\ ( X = F /\ Y = G ) ) ) |