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 |
|
funcestrcsetc.g |
|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) ) |
8 |
1 2 3 4 5 6 7
|
funcestrcsetc |
|- ( ph -> F ( E Func S ) G ) |
9 |
1 2 3 4 5 6 7
|
funcestrcsetclem8 |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a G b ) : ( a ( Hom ` E ) b ) --> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) |
10 |
5
|
adantr |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> U e. WUni ) |
11 |
|
eqid |
|- ( Hom ` E ) = ( Hom ` E ) |
12 |
1 5
|
estrcbas |
|- ( ph -> U = ( Base ` E ) ) |
13 |
3 12
|
eqtr4id |
|- ( ph -> B = U ) |
14 |
13
|
eleq2d |
|- ( ph -> ( a e. B <-> a e. U ) ) |
15 |
14
|
biimpcd |
|- ( a e. B -> ( ph -> a e. U ) ) |
16 |
15
|
adantr |
|- ( ( a e. B /\ b e. B ) -> ( ph -> a e. U ) ) |
17 |
16
|
impcom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> a e. U ) |
18 |
13
|
eleq2d |
|- ( ph -> ( b e. B <-> b e. U ) ) |
19 |
18
|
biimpcd |
|- ( b e. B -> ( ph -> b e. U ) ) |
20 |
19
|
adantl |
|- ( ( a e. B /\ b e. B ) -> ( ph -> b e. U ) ) |
21 |
20
|
impcom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> b e. U ) |
22 |
|
eqid |
|- ( Base ` a ) = ( Base ` a ) |
23 |
|
eqid |
|- ( Base ` b ) = ( Base ` b ) |
24 |
1 10 11 17 21 22 23
|
estrchom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( Hom ` E ) b ) = ( ( Base ` b ) ^m ( Base ` a ) ) ) |
25 |
24
|
eleq2d |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( a ( Hom ` E ) b ) <-> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) ) |
26 |
1 2 3 4 5 6 7 22 23
|
funcestrcsetclem6 |
|- ( ( ph /\ ( a e. B /\ b e. B ) /\ h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) -> ( ( a G b ) ` h ) = h ) |
27 |
26
|
3expia |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) -> ( ( a G b ) ` h ) = h ) ) |
28 |
25 27
|
sylbid |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( a ( Hom ` E ) b ) -> ( ( a G b ) ` h ) = h ) ) |
29 |
28
|
com12 |
|- ( h e. ( a ( Hom ` E ) b ) -> ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( ( a G b ) ` h ) = h ) ) |
30 |
29
|
adantr |
|- ( ( h e. ( a ( Hom ` E ) b ) /\ k e. ( a ( Hom ` E ) b ) ) -> ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( ( a G b ) ` h ) = h ) ) |
31 |
30
|
impcom |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ ( h e. ( a ( Hom ` E ) b ) /\ k e. ( a ( Hom ` E ) b ) ) ) -> ( ( a G b ) ` h ) = h ) |
32 |
24
|
eleq2d |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( k e. ( a ( Hom ` E ) b ) <-> k e. ( ( Base ` b ) ^m ( Base ` a ) ) ) ) |
33 |
1 2 3 4 5 6 7 22 23
|
funcestrcsetclem6 |
|- ( ( ph /\ ( a e. B /\ b e. B ) /\ k e. ( ( Base ` b ) ^m ( Base ` a ) ) ) -> ( ( a G b ) ` k ) = k ) |
34 |
33
|
3expia |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( k e. ( ( Base ` b ) ^m ( Base ` a ) ) -> ( ( a G b ) ` k ) = k ) ) |
35 |
32 34
|
sylbid |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( k e. ( a ( Hom ` E ) b ) -> ( ( a G b ) ` k ) = k ) ) |
36 |
35
|
com12 |
|- ( k e. ( a ( Hom ` E ) b ) -> ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( ( a G b ) ` k ) = k ) ) |
37 |
36
|
adantl |
|- ( ( h e. ( a ( Hom ` E ) b ) /\ k e. ( a ( Hom ` E ) b ) ) -> ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( ( a G b ) ` k ) = k ) ) |
38 |
37
|
impcom |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ ( h e. ( a ( Hom ` E ) b ) /\ k e. ( a ( Hom ` E ) b ) ) ) -> ( ( a G b ) ` k ) = k ) |
39 |
31 38
|
eqeq12d |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ ( h e. ( a ( Hom ` E ) b ) /\ k e. ( a ( Hom ` E ) b ) ) ) -> ( ( ( a G b ) ` h ) = ( ( a G b ) ` k ) <-> h = k ) ) |
40 |
39
|
biimpd |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ ( h e. ( a ( Hom ` E ) b ) /\ k e. ( a ( Hom ` E ) b ) ) ) -> ( ( ( a G b ) ` h ) = ( ( a G b ) ` k ) -> h = k ) ) |
41 |
40
|
ralrimivva |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> A. h e. ( a ( Hom ` E ) b ) A. k e. ( a ( Hom ` E ) b ) ( ( ( a G b ) ` h ) = ( ( a G b ) ` k ) -> h = k ) ) |
42 |
|
dff13 |
|- ( ( a G b ) : ( a ( Hom ` E ) b ) -1-1-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) <-> ( ( a G b ) : ( a ( Hom ` E ) b ) --> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) /\ A. h e. ( a ( Hom ` E ) b ) A. k e. ( a ( Hom ` E ) b ) ( ( ( a G b ) ` h ) = ( ( a G b ) ` k ) -> h = k ) ) ) |
43 |
9 41 42
|
sylanbrc |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a G b ) : ( a ( Hom ` E ) b ) -1-1-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) |
44 |
43
|
ralrimivva |
|- ( ph -> A. a e. B A. b e. B ( a G b ) : ( a ( Hom ` E ) b ) -1-1-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) |
45 |
|
eqid |
|- ( Hom ` S ) = ( Hom ` S ) |
46 |
3 11 45
|
isfth2 |
|- ( F ( E Faith S ) G <-> ( F ( E Func S ) G /\ A. a e. B A. b e. B ( a G b ) : ( a ( Hom ` E ) b ) -1-1-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) ) |
47 |
8 44 46
|
sylanbrc |
|- ( ph -> F ( E Faith S ) G ) |