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 ` S ) = ( Hom ` S ) |
12 |
1 2 3 4 5 6
|
funcestrcsetclem2 |
|- ( ( ph /\ a e. B ) -> ( F ` a ) e. U ) |
13 |
12
|
adantrr |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( F ` a ) e. U ) |
14 |
1 2 3 4 5 6
|
funcestrcsetclem2 |
|- ( ( ph /\ b e. B ) -> ( F ` b ) e. U ) |
15 |
14
|
adantrl |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( F ` b ) e. U ) |
16 |
2 10 11 13 15
|
elsetchom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) <-> h : ( F ` a ) --> ( F ` b ) ) ) |
17 |
1 2 3 4 5 6
|
funcestrcsetclem1 |
|- ( ( ph /\ a e. B ) -> ( F ` a ) = ( Base ` a ) ) |
18 |
17
|
adantrr |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( F ` a ) = ( Base ` a ) ) |
19 |
1 2 3 4 5 6
|
funcestrcsetclem1 |
|- ( ( ph /\ b e. B ) -> ( F ` b ) = ( Base ` b ) ) |
20 |
19
|
adantrl |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( F ` b ) = ( Base ` b ) ) |
21 |
18 20
|
feq23d |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h : ( F ` a ) --> ( F ` b ) <-> h : ( Base ` a ) --> ( Base ` b ) ) ) |
22 |
16 21
|
bitrd |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) ) |
23 |
|
fvex |
|- ( Base ` b ) e. _V |
24 |
|
fvex |
|- ( Base ` a ) e. _V |
25 |
23 24
|
pm3.2i |
|- ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V ) |
26 |
|
elmapg |
|- ( ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) ) |
27 |
25 26
|
mp1i |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) ) |
28 |
27
|
biimpar |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) |
29 |
|
equequ2 |
|- ( k = h -> ( h = k <-> h = h ) ) |
30 |
29
|
adantl |
|- ( ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) /\ k = h ) -> ( h = k <-> h = h ) ) |
31 |
|
eqidd |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h = h ) |
32 |
28 30 31
|
rspcedvd |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = k ) |
33 |
|
eqid |
|- ( Base ` a ) = ( Base ` a ) |
34 |
|
eqid |
|- ( Base ` b ) = ( Base ` b ) |
35 |
1 2 3 4 5 6 7 33 34
|
funcestrcsetclem6 |
|- ( ( ph /\ ( a e. B /\ b e. B ) /\ k e. ( ( Base ` b ) ^m ( Base ` a ) ) ) -> ( ( a G b ) ` k ) = k ) |
36 |
35
|
3expa |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ k e. ( ( Base ` b ) ^m ( Base ` a ) ) ) -> ( ( a G b ) ` k ) = k ) |
37 |
36
|
eqeq2d |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ k e. ( ( Base ` b ) ^m ( Base ` a ) ) ) -> ( h = ( ( a G b ) ` k ) <-> h = k ) ) |
38 |
37
|
rexbidva |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = ( ( a G b ) ` k ) <-> E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = k ) ) |
39 |
38
|
adantr |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> ( E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = ( ( a G b ) ` k ) <-> E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = k ) ) |
40 |
32 39
|
mpbird |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = ( ( a G b ) ` k ) ) |
41 |
|
eqid |
|- ( Hom ` E ) = ( Hom ` E ) |
42 |
1 5
|
estrcbas |
|- ( ph -> U = ( Base ` E ) ) |
43 |
3 42
|
eqtr4id |
|- ( ph -> B = U ) |
44 |
43
|
eleq2d |
|- ( ph -> ( a e. B <-> a e. U ) ) |
45 |
44
|
biimpcd |
|- ( a e. B -> ( ph -> a e. U ) ) |
46 |
45
|
adantr |
|- ( ( a e. B /\ b e. B ) -> ( ph -> a e. U ) ) |
47 |
46
|
impcom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> a e. U ) |
48 |
43
|
eleq2d |
|- ( ph -> ( b e. B <-> b e. U ) ) |
49 |
48
|
biimpcd |
|- ( b e. B -> ( ph -> b e. U ) ) |
50 |
49
|
adantl |
|- ( ( a e. B /\ b e. B ) -> ( ph -> b e. U ) ) |
51 |
50
|
impcom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> b e. U ) |
52 |
1 10 41 47 51 33 34
|
estrchom |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( Hom ` E ) b ) = ( ( Base ` b ) ^m ( Base ` a ) ) ) |
53 |
52
|
rexeqdv |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) <-> E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = ( ( a G b ) ` k ) ) ) |
54 |
53
|
adantr |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> ( E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) <-> E. k e. ( ( Base ` b ) ^m ( Base ` a ) ) h = ( ( a G b ) ` k ) ) ) |
55 |
40 54
|
mpbird |
|- ( ( ( ph /\ ( a e. B /\ b e. B ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) ) |
56 |
55
|
ex |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h : ( Base ` a ) --> ( Base ` b ) -> E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) ) ) |
57 |
22 56
|
sylbid |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( h e. ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) -> E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) ) ) |
58 |
57
|
ralrimiv |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> A. h e. ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) ) |
59 |
|
dffo3 |
|- ( ( a G b ) : ( a ( Hom ` E ) b ) -onto-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) <-> ( ( a G b ) : ( a ( Hom ` E ) b ) --> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) /\ A. h e. ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) E. k e. ( a ( Hom ` E ) b ) h = ( ( a G b ) ` k ) ) ) |
60 |
9 58 59
|
sylanbrc |
|- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a G b ) : ( a ( Hom ` E ) b ) -onto-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) |
61 |
60
|
ralrimivva |
|- ( ph -> A. a e. B A. b e. B ( a G b ) : ( a ( Hom ` E ) b ) -onto-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) |
62 |
3 11 41
|
isfull2 |
|- ( F ( E Full S ) G <-> ( F ( E Func S ) G /\ A. a e. B A. b e. B ( a G b ) : ( a ( Hom ` E ) b ) -onto-> ( ( F ` a ) ( Hom ` S ) ( F ` b ) ) ) ) |
63 |
8 61 62
|
sylanbrc |
|- ( ph -> F ( E Full S ) G ) |