| Step |
Hyp |
Ref |
Expression |
| 1 |
|
funcsetcestrc.s |
|- S = ( SetCat ` U ) |
| 2 |
|
funcsetcestrc.c |
|- C = ( Base ` S ) |
| 3 |
|
funcsetcestrc.f |
|- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) ) |
| 4 |
|
funcsetcestrc.u |
|- ( ph -> U e. WUni ) |
| 5 |
|
funcsetcestrc.o |
|- ( ph -> _om e. U ) |
| 6 |
|
funcsetcestrclem3.e |
|- E = ( ExtStrCat ` U ) |
| 7 |
|
funcsetcestrclem3.b |
|- B = ( Base ` E ) |
| 8 |
1 2 4 5
|
setc1strwun |
|- ( ( ph /\ x e. C ) -> { <. ( Base ` ndx ) , x >. } e. U ) |
| 9 |
6 4
|
estrcbas |
|- ( ph -> U = ( Base ` E ) ) |
| 10 |
9
|
eqcomd |
|- ( ph -> ( Base ` E ) = U ) |
| 11 |
10
|
adantr |
|- ( ( ph /\ x e. C ) -> ( Base ` E ) = U ) |
| 12 |
8 11
|
eleqtrrd |
|- ( ( ph /\ x e. C ) -> { <. ( Base ` ndx ) , x >. } e. ( Base ` E ) ) |
| 13 |
12 7
|
eleqtrrdi |
|- ( ( ph /\ x e. C ) -> { <. ( Base ` ndx ) , x >. } e. B ) |
| 14 |
3 13
|
fmpt3d |
|- ( ph -> F : C --> B ) |