Step |
Hyp |
Ref |
Expression |
1 |
|
fprlem.1 |
|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) } |
2 |
|
fprlem.2 |
|- F = frecs ( R , A , G ) |
3 |
|
vex |
|- x e. _V |
4 |
|
vex |
|- u e. _V |
5 |
3 4
|
breldm |
|- ( x g u -> x e. dom g ) |
6 |
|
vex |
|- v e. _V |
7 |
3 6
|
breldm |
|- ( x h v -> x e. dom h ) |
8 |
|
elin |
|- ( x e. ( dom g i^i dom h ) <-> ( x e. dom g /\ x e. dom h ) ) |
9 |
8
|
biimpri |
|- ( ( x e. dom g /\ x e. dom h ) -> x e. ( dom g i^i dom h ) ) |
10 |
5 7 9
|
syl2an |
|- ( ( x g u /\ x h v ) -> x e. ( dom g i^i dom h ) ) |
11 |
|
id |
|- ( ( x g u /\ x h v ) -> ( x g u /\ x h v ) ) |
12 |
4
|
brresi |
|- ( x ( g |` ( dom g i^i dom h ) ) u <-> ( x e. ( dom g i^i dom h ) /\ x g u ) ) |
13 |
6
|
brresi |
|- ( x ( h |` ( dom g i^i dom h ) ) v <-> ( x e. ( dom g i^i dom h ) /\ x h v ) ) |
14 |
12 13
|
anbi12i |
|- ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) <-> ( ( x e. ( dom g i^i dom h ) /\ x g u ) /\ ( x e. ( dom g i^i dom h ) /\ x h v ) ) ) |
15 |
|
an4 |
|- ( ( ( x e. ( dom g i^i dom h ) /\ x g u ) /\ ( x e. ( dom g i^i dom h ) /\ x h v ) ) <-> ( ( x e. ( dom g i^i dom h ) /\ x e. ( dom g i^i dom h ) ) /\ ( x g u /\ x h v ) ) ) |
16 |
14 15
|
bitri |
|- ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) <-> ( ( x e. ( dom g i^i dom h ) /\ x e. ( dom g i^i dom h ) ) /\ ( x g u /\ x h v ) ) ) |
17 |
10 10 11 16
|
syl21anbrc |
|- ( ( x g u /\ x h v ) -> ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) ) |
18 |
|
inss2 |
|- ( dom g i^i dom h ) C_ dom h |
19 |
1
|
frrlem3 |
|- ( h e. B -> dom h C_ A ) |
20 |
18 19
|
sstrid |
|- ( h e. B -> ( dom g i^i dom h ) C_ A ) |
21 |
20
|
adantl |
|- ( ( g e. B /\ h e. B ) -> ( dom g i^i dom h ) C_ A ) |
22 |
21
|
adantl |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( dom g i^i dom h ) C_ A ) |
23 |
|
simpl1 |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> R Fr A ) |
24 |
|
frss |
|- ( ( dom g i^i dom h ) C_ A -> ( R Fr A -> R Fr ( dom g i^i dom h ) ) ) |
25 |
22 23 24
|
sylc |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> R Fr ( dom g i^i dom h ) ) |
26 |
|
simpl2 |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> R Po A ) |
27 |
|
poss |
|- ( ( dom g i^i dom h ) C_ A -> ( R Po A -> R Po ( dom g i^i dom h ) ) ) |
28 |
22 26 27
|
sylc |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> R Po ( dom g i^i dom h ) ) |
29 |
|
simpl3 |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> R Se A ) |
30 |
|
sess2 |
|- ( ( dom g i^i dom h ) C_ A -> ( R Se A -> R Se ( dom g i^i dom h ) ) ) |
31 |
22 29 30
|
sylc |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> R Se ( dom g i^i dom h ) ) |
32 |
1
|
frrlem4 |
|- ( ( g e. B /\ h e. B ) -> ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( g |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) |
33 |
32
|
adantl |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( g |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) |
34 |
1
|
frrlem4 |
|- ( ( h e. B /\ g e. B ) -> ( ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) /\ A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( a G ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) ) |
35 |
|
incom |
|- ( dom g i^i dom h ) = ( dom h i^i dom g ) |
36 |
35
|
reseq2i |
|- ( h |` ( dom g i^i dom h ) ) = ( h |` ( dom h i^i dom g ) ) |
37 |
|
fneq12 |
|- ( ( ( h |` ( dom g i^i dom h ) ) = ( h |` ( dom h i^i dom g ) ) /\ ( dom g i^i dom h ) = ( dom h i^i dom g ) ) -> ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) <-> ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) ) ) |
38 |
36 35 37
|
mp2an |
|- ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) <-> ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) ) |
39 |
36
|
fveq1i |
|- ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( ( h |` ( dom h i^i dom g ) ) ` a ) |
40 |
|
predeq2 |
|- ( ( dom g i^i dom h ) = ( dom h i^i dom g ) -> Pred ( R , ( dom g i^i dom h ) , a ) = Pred ( R , ( dom h i^i dom g ) , a ) ) |
41 |
35 40
|
ax-mp |
|- Pred ( R , ( dom g i^i dom h ) , a ) = Pred ( R , ( dom h i^i dom g ) , a ) |
42 |
36 41
|
reseq12i |
|- ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) = ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) |
43 |
42
|
oveq2i |
|- ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) = ( a G ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) |
44 |
39 43
|
eqeq12i |
|- ( ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) <-> ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( a G ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) |
45 |
35 44
|
raleqbii |
|- ( A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) <-> A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( a G ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) |
46 |
38 45
|
anbi12i |
|- ( ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) <-> ( ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) /\ A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( a G ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) ) |
47 |
34 46
|
sylibr |
|- ( ( h e. B /\ g e. B ) -> ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) |
48 |
47
|
ancoms |
|- ( ( g e. B /\ h e. B ) -> ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) |
49 |
48
|
adantl |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) |
50 |
|
fpr3g |
|- ( ( ( R Fr ( dom g i^i dom h ) /\ R Po ( dom g i^i dom h ) /\ R Se ( dom g i^i dom h ) ) /\ ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( g |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) /\ ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( a G ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) -> ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) ) |
51 |
25 28 31 33 49 50
|
syl311anc |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) ) |
52 |
51
|
breqd |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( x ( g |` ( dom g i^i dom h ) ) v <-> x ( h |` ( dom g i^i dom h ) ) v ) ) |
53 |
52
|
biimprd |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( x ( h |` ( dom g i^i dom h ) ) v -> x ( g |` ( dom g i^i dom h ) ) v ) ) |
54 |
1
|
frrlem2 |
|- ( g e. B -> Fun g ) |
55 |
54
|
ad2antrl |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> Fun g ) |
56 |
|
funres |
|- ( Fun g -> Fun ( g |` ( dom g i^i dom h ) ) ) |
57 |
|
dffun2 |
|- ( Fun ( g |` ( dom g i^i dom h ) ) <-> ( Rel ( g |` ( dom g i^i dom h ) ) /\ A. x A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) ) ) |
58 |
|
2sp |
|- ( A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) ) |
59 |
58
|
sps |
|- ( A. x A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) ) |
60 |
57 59
|
simplbiim |
|- ( Fun ( g |` ( dom g i^i dom h ) ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) ) |
61 |
55 56 60
|
3syl |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) ) |
62 |
53 61
|
sylan2d |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) -> u = v ) ) |
63 |
17 62
|
syl5 |
|- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( g e. B /\ h e. B ) ) -> ( ( x g u /\ x h v ) -> u = v ) ) |