| Step |
Hyp |
Ref |
Expression |
| 1 |
|
funcestrcsetc.e |
|- E = ( ExtStrCat ` U ) |
| 2 |
|
funcestrcsetc.s |
|- S = ( SetCat ` U ) |
| 3 |
|
funcestrcsetc.b |
|- B = ( Base ` E ) |
| 4 |
|
funcestrcsetc.c |
|- C = ( Base ` S ) |
| 5 |
|
funcestrcsetc.u |
|- ( ph -> U e. WUni ) |
| 6 |
|
funcestrcsetc.f |
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) ) |
| 7 |
1 3 5
|
estrcbasbas |
|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. U ) |
| 8 |
2 5
|
setcbas |
|- ( ph -> U = ( Base ` S ) ) |
| 9 |
8
|
eqcomd |
|- ( ph -> ( Base ` S ) = U ) |
| 10 |
9
|
adantr |
|- ( ( ph /\ x e. B ) -> ( Base ` S ) = U ) |
| 11 |
7 10
|
eleqtrrd |
|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. ( Base ` S ) ) |
| 12 |
11 4
|
eleqtrrdi |
|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. C ) |
| 13 |
6 12
|
fmpt3d |
|- ( ph -> F : B --> C ) |