Step |
Hyp |
Ref |
Expression |
1 |
|
frrlem4.1 |
|- 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 |
1
|
frrlem2 |
|- ( g e. B -> Fun g ) |
3 |
2
|
funfnd |
|- ( g e. B -> g Fn dom g ) |
4 |
|
fnresin1 |
|- ( g Fn dom g -> ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) ) |
5 |
3 4
|
syl |
|- ( g e. B -> ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) ) |
6 |
5
|
adantr |
|- ( ( g e. B /\ h e. B ) -> ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) ) |
7 |
1
|
frrlem1 |
|- B = { g | E. b ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) } |
8 |
7
|
abeq2i |
|- ( g e. B <-> E. b ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) ) |
9 |
|
fndm |
|- ( g Fn b -> dom g = b ) |
10 |
9
|
adantr |
|- ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) ) -> dom g = b ) |
11 |
10
|
raleqdv |
|- ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) ) -> ( A. a e. dom g ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) <-> A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) ) |
12 |
11
|
biimp3ar |
|- ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) -> A. a e. dom g ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) |
13 |
|
rsp |
|- ( A. a e. dom g ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) -> ( a e. dom g -> ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) ) |
14 |
12 13
|
syl |
|- ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) -> ( a e. dom g -> ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) ) |
15 |
14
|
exlimiv |
|- ( E. b ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) -> ( a e. dom g -> ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) ) |
16 |
8 15
|
sylbi |
|- ( g e. B -> ( a e. dom g -> ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) ) |
17 |
|
elinel1 |
|- ( a e. ( dom g i^i dom h ) -> a e. dom g ) |
18 |
16 17
|
impel |
|- ( ( g e. B /\ a e. ( dom g i^i dom h ) ) -> ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) |
19 |
18
|
adantlr |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) |
20 |
|
simpr |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> a e. ( dom g i^i dom h ) ) |
21 |
20
|
fvresd |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( ( g |` ( dom g i^i dom h ) ) ` a ) = ( g ` a ) ) |
22 |
|
resres |
|- ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) = ( g |` ( ( dom g i^i dom h ) i^i Pred ( R , ( dom g i^i dom h ) , a ) ) ) |
23 |
|
predss |
|- Pred ( R , ( dom g i^i dom h ) , a ) C_ ( dom g i^i dom h ) |
24 |
|
sseqin2 |
|- ( Pred ( R , ( dom g i^i dom h ) , a ) C_ ( dom g i^i dom h ) <-> ( ( dom g i^i dom h ) i^i Pred ( R , ( dom g i^i dom h ) , a ) ) = Pred ( R , ( dom g i^i dom h ) , a ) ) |
25 |
23 24
|
mpbi |
|- ( ( dom g i^i dom h ) i^i Pred ( R , ( dom g i^i dom h ) , a ) ) = Pred ( R , ( dom g i^i dom h ) , a ) |
26 |
1
|
frrlem1 |
|- B = { h | E. c ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) } |
27 |
26
|
abeq2i |
|- ( h e. B <-> E. c ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) |
28 |
|
exdistrv |
|- ( E. b E. c ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) <-> ( E. b ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ E. c ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) ) |
29 |
|
inss1 |
|- ( b i^i c ) C_ b |
30 |
|
simpl2l |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> b C_ A ) |
31 |
29 30
|
sstrid |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> ( b i^i c ) C_ A ) |
32 |
|
simp2r |
|- ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) -> A. a e. b Pred ( R , A , a ) C_ b ) |
33 |
|
simp2r |
|- ( ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) -> A. a e. c Pred ( R , A , a ) C_ c ) |
34 |
|
nfra1 |
|- F/ a A. a e. b Pred ( R , A , a ) C_ b |
35 |
|
nfra1 |
|- F/ a A. a e. c Pred ( R , A , a ) C_ c |
36 |
34 35
|
nfan |
|- F/ a ( A. a e. b Pred ( R , A , a ) C_ b /\ A. a e. c Pred ( R , A , a ) C_ c ) |
37 |
|
elinel1 |
|- ( a e. ( b i^i c ) -> a e. b ) |
38 |
|
rsp |
|- ( A. a e. b Pred ( R , A , a ) C_ b -> ( a e. b -> Pred ( R , A , a ) C_ b ) ) |
39 |
37 38
|
syl5com |
|- ( a e. ( b i^i c ) -> ( A. a e. b Pred ( R , A , a ) C_ b -> Pred ( R , A , a ) C_ b ) ) |
40 |
|
elinel2 |
|- ( a e. ( b i^i c ) -> a e. c ) |
41 |
|
rsp |
|- ( A. a e. c Pred ( R , A , a ) C_ c -> ( a e. c -> Pred ( R , A , a ) C_ c ) ) |
42 |
40 41
|
syl5com |
|- ( a e. ( b i^i c ) -> ( A. a e. c Pred ( R , A , a ) C_ c -> Pred ( R , A , a ) C_ c ) ) |
43 |
39 42
|
anim12d |
|- ( a e. ( b i^i c ) -> ( ( A. a e. b Pred ( R , A , a ) C_ b /\ A. a e. c Pred ( R , A , a ) C_ c ) -> ( Pred ( R , A , a ) C_ b /\ Pred ( R , A , a ) C_ c ) ) ) |
44 |
|
ssin |
|- ( ( Pred ( R , A , a ) C_ b /\ Pred ( R , A , a ) C_ c ) <-> Pred ( R , A , a ) C_ ( b i^i c ) ) |
45 |
44
|
biimpi |
|- ( ( Pred ( R , A , a ) C_ b /\ Pred ( R , A , a ) C_ c ) -> Pred ( R , A , a ) C_ ( b i^i c ) ) |
46 |
43 45
|
syl6com |
|- ( ( A. a e. b Pred ( R , A , a ) C_ b /\ A. a e. c Pred ( R , A , a ) C_ c ) -> ( a e. ( b i^i c ) -> Pred ( R , A , a ) C_ ( b i^i c ) ) ) |
47 |
36 46
|
ralrimi |
|- ( ( A. a e. b Pred ( R , A , a ) C_ b /\ A. a e. c Pred ( R , A , a ) C_ c ) -> A. a e. ( b i^i c ) Pred ( R , A , a ) C_ ( b i^i c ) ) |
48 |
32 33 47
|
syl2an |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> A. a e. ( b i^i c ) Pred ( R , A , a ) C_ ( b i^i c ) ) |
49 |
|
simpl1 |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> g Fn b ) |
50 |
49
|
fndmd |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> dom g = b ) |
51 |
|
simpr1 |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> h Fn c ) |
52 |
51
|
fndmd |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> dom h = c ) |
53 |
|
ineq12 |
|- ( ( dom g = b /\ dom h = c ) -> ( dom g i^i dom h ) = ( b i^i c ) ) |
54 |
53
|
sseq1d |
|- ( ( dom g = b /\ dom h = c ) -> ( ( dom g i^i dom h ) C_ A <-> ( b i^i c ) C_ A ) ) |
55 |
53
|
sseq2d |
|- ( ( dom g = b /\ dom h = c ) -> ( Pred ( R , A , a ) C_ ( dom g i^i dom h ) <-> Pred ( R , A , a ) C_ ( b i^i c ) ) ) |
56 |
53 55
|
raleqbidv |
|- ( ( dom g = b /\ dom h = c ) -> ( A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) <-> A. a e. ( b i^i c ) Pred ( R , A , a ) C_ ( b i^i c ) ) ) |
57 |
54 56
|
anbi12d |
|- ( ( dom g = b /\ dom h = c ) -> ( ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) <-> ( ( b i^i c ) C_ A /\ A. a e. ( b i^i c ) Pred ( R , A , a ) C_ ( b i^i c ) ) ) ) |
58 |
50 52 57
|
syl2anc |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> ( ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) <-> ( ( b i^i c ) C_ A /\ A. a e. ( b i^i c ) Pred ( R , A , a ) C_ ( b i^i c ) ) ) ) |
59 |
31 48 58
|
mpbir2and |
|- ( ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) ) |
60 |
59
|
exlimivv |
|- ( E. b E. c ( ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) ) |
61 |
28 60
|
sylbir |
|- ( ( E. b ( g Fn b /\ ( b C_ A /\ A. a e. b Pred ( R , A , a ) C_ b ) /\ A. a e. b ( g ` a ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) /\ E. c ( h Fn c /\ ( c C_ A /\ A. a e. c Pred ( R , A , a ) C_ c ) /\ A. a e. c ( h ` a ) = ( a G ( h |` Pred ( R , A , a ) ) ) ) ) -> ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) ) |
62 |
8 27 61
|
syl2anb |
|- ( ( g e. B /\ h e. B ) -> ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) ) |
63 |
62
|
adantr |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) ) |
64 |
|
preddowncl |
|- ( ( ( dom g i^i dom h ) C_ A /\ A. a e. ( dom g i^i dom h ) Pred ( R , A , a ) C_ ( dom g i^i dom h ) ) -> ( a e. ( dom g i^i dom h ) -> Pred ( R , ( dom g i^i dom h ) , a ) = Pred ( R , A , a ) ) ) |
65 |
63 20 64
|
sylc |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> Pred ( R , ( dom g i^i dom h ) , a ) = Pred ( R , A , a ) ) |
66 |
25 65
|
eqtrid |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( ( dom g i^i dom h ) i^i Pred ( R , ( dom g i^i dom h ) , a ) ) = Pred ( R , A , a ) ) |
67 |
66
|
reseq2d |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( g |` ( ( dom g i^i dom h ) i^i Pred ( R , ( dom g i^i dom h ) , a ) ) ) = ( g |` Pred ( R , A , a ) ) ) |
68 |
22 67
|
eqtrid |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) = ( g |` Pred ( R , A , a ) ) ) |
69 |
68
|
oveq2d |
|- ( ( ( g e. B /\ h e. B ) /\ a e. ( dom g i^i dom h ) ) -> ( a G ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) = ( a G ( g |` Pred ( R , A , a ) ) ) ) |
70 |
19 21 69
|
3eqtr4d |
|- ( ( ( g e. B /\ h e. B ) /\ 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 ) ) ) ) |
71 |
70
|
ralrimiva |
|- ( ( g e. B /\ h e. B ) -> 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 ) ) ) ) |
72 |
6 71
|
jca |
|- ( ( 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 ) ) ) ) ) |