Step |
Hyp |
Ref |
Expression |
1 |
|
frrlem11.1 |
|- 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 |
|
frrlem11.2 |
|- F = frecs ( R , A , G ) |
3 |
|
frrlem11.3 |
|- ( ( ph /\ ( g e. B /\ h e. B ) ) -> ( ( x g u /\ x h v ) -> u = v ) ) |
4 |
|
frrlem11.4 |
|- C = ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) |
5 |
|
frrlem12.5 |
|- ( ph -> R Fr A ) |
6 |
|
frrlem12.6 |
|- ( ( ph /\ z e. A ) -> Pred ( R , A , z ) C_ S ) |
7 |
|
frrlem12.7 |
|- ( ( ph /\ z e. A ) -> A. w e. S Pred ( R , A , w ) C_ S ) |
8 |
|
frrlem13.8 |
|- ( ( ph /\ z e. A ) -> S e. _V ) |
9 |
|
frrlem13.9 |
|- ( ( ph /\ z e. A ) -> S C_ A ) |
10 |
|
eldifi |
|- ( z e. ( A \ dom F ) -> z e. A ) |
11 |
10 8
|
sylan2 |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> S e. _V ) |
12 |
11
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> S e. _V ) |
13 |
|
inex1g |
|- ( S e. _V -> ( S i^i dom F ) e. _V ) |
14 |
|
snex |
|- { z } e. _V |
15 |
|
unexg |
|- ( ( ( S i^i dom F ) e. _V /\ { z } e. _V ) -> ( ( S i^i dom F ) u. { z } ) e. _V ) |
16 |
13 14 15
|
sylancl |
|- ( S e. _V -> ( ( S i^i dom F ) u. { z } ) e. _V ) |
17 |
12 16
|
syl |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( S i^i dom F ) u. { z } ) e. _V ) |
18 |
1 2 3 4
|
frrlem11 |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> C Fn ( ( S i^i dom F ) u. { z } ) ) |
19 |
18
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> C Fn ( ( S i^i dom F ) u. { z } ) ) |
20 |
|
inss1 |
|- ( S i^i dom F ) C_ S |
21 |
10 9
|
sylan2 |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> S C_ A ) |
22 |
21
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> S C_ A ) |
23 |
20 22
|
sstrid |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( S i^i dom F ) C_ A ) |
24 |
10
|
adantl |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> z e. A ) |
25 |
24
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> z e. A ) |
26 |
25
|
snssd |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> { z } C_ A ) |
27 |
23 26
|
unssd |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( S i^i dom F ) u. { z } ) C_ A ) |
28 |
|
elun |
|- ( w e. ( ( S i^i dom F ) u. { z } ) <-> ( w e. ( S i^i dom F ) \/ w e. { z } ) ) |
29 |
|
elin |
|- ( w e. ( S i^i dom F ) <-> ( w e. S /\ w e. dom F ) ) |
30 |
|
velsn |
|- ( w e. { z } <-> w = z ) |
31 |
29 30
|
orbi12i |
|- ( ( w e. ( S i^i dom F ) \/ w e. { z } ) <-> ( ( w e. S /\ w e. dom F ) \/ w = z ) ) |
32 |
28 31
|
bitri |
|- ( w e. ( ( S i^i dom F ) u. { z } ) <-> ( ( w e. S /\ w e. dom F ) \/ w = z ) ) |
33 |
10 7
|
sylan2 |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> A. w e. S Pred ( R , A , w ) C_ S ) |
34 |
33
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> A. w e. S Pred ( R , A , w ) C_ S ) |
35 |
|
rsp |
|- ( A. w e. S Pred ( R , A , w ) C_ S -> ( w e. S -> Pred ( R , A , w ) C_ S ) ) |
36 |
34 35
|
syl |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( w e. S -> Pred ( R , A , w ) C_ S ) ) |
37 |
1 2
|
frrlem8 |
|- ( w e. dom F -> Pred ( R , A , w ) C_ dom F ) |
38 |
36 37
|
anim12d1 |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( w e. S /\ w e. dom F ) -> ( Pred ( R , A , w ) C_ S /\ Pred ( R , A , w ) C_ dom F ) ) ) |
39 |
|
ssin |
|- ( ( Pred ( R , A , w ) C_ S /\ Pred ( R , A , w ) C_ dom F ) <-> Pred ( R , A , w ) C_ ( S i^i dom F ) ) |
40 |
38 39
|
syl6ib |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( w e. S /\ w e. dom F ) -> Pred ( R , A , w ) C_ ( S i^i dom F ) ) ) |
41 |
10 6
|
sylan2 |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> Pred ( R , A , z ) C_ S ) |
42 |
41
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> Pred ( R , A , z ) C_ S ) |
43 |
|
preddif |
|- Pred ( R , ( A \ dom F ) , z ) = ( Pred ( R , A , z ) \ Pred ( R , dom F , z ) ) |
44 |
43
|
eqeq1i |
|- ( Pred ( R , ( A \ dom F ) , z ) = (/) <-> ( Pred ( R , A , z ) \ Pred ( R , dom F , z ) ) = (/) ) |
45 |
|
ssdif0 |
|- ( Pred ( R , A , z ) C_ Pred ( R , dom F , z ) <-> ( Pred ( R , A , z ) \ Pred ( R , dom F , z ) ) = (/) ) |
46 |
44 45
|
sylbb2 |
|- ( Pred ( R , ( A \ dom F ) , z ) = (/) -> Pred ( R , A , z ) C_ Pred ( R , dom F , z ) ) |
47 |
|
predss |
|- Pred ( R , dom F , z ) C_ dom F |
48 |
46 47
|
sstrdi |
|- ( Pred ( R , ( A \ dom F ) , z ) = (/) -> Pred ( R , A , z ) C_ dom F ) |
49 |
48
|
adantl |
|- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> Pred ( R , A , z ) C_ dom F ) |
50 |
49
|
adantl |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> Pred ( R , A , z ) C_ dom F ) |
51 |
42 50
|
ssind |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> Pred ( R , A , z ) C_ ( S i^i dom F ) ) |
52 |
|
predeq3 |
|- ( w = z -> Pred ( R , A , w ) = Pred ( R , A , z ) ) |
53 |
52
|
sseq1d |
|- ( w = z -> ( Pred ( R , A , w ) C_ ( S i^i dom F ) <-> Pred ( R , A , z ) C_ ( S i^i dom F ) ) ) |
54 |
51 53
|
syl5ibrcom |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( w = z -> Pred ( R , A , w ) C_ ( S i^i dom F ) ) ) |
55 |
40 54
|
jaod |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( ( w e. S /\ w e. dom F ) \/ w = z ) -> Pred ( R , A , w ) C_ ( S i^i dom F ) ) ) |
56 |
32 55
|
syl5bi |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( w e. ( ( S i^i dom F ) u. { z } ) -> Pred ( R , A , w ) C_ ( S i^i dom F ) ) ) |
57 |
56
|
imp |
|- ( ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) /\ w e. ( ( S i^i dom F ) u. { z } ) ) -> Pred ( R , A , w ) C_ ( S i^i dom F ) ) |
58 |
|
ssun1 |
|- ( S i^i dom F ) C_ ( ( S i^i dom F ) u. { z } ) |
59 |
57 58
|
sstrdi |
|- ( ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) /\ w e. ( ( S i^i dom F ) u. { z } ) ) -> Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) |
60 |
59
|
ralrimiva |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) |
61 |
27 60
|
jca |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( ( S i^i dom F ) u. { z } ) C_ A /\ A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) ) |
62 |
1 2 3 4 5 6 7
|
frrlem12 |
|- ( ( ph /\ z e. ( A \ dom F ) /\ w e. ( ( S i^i dom F ) u. { z } ) ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) |
63 |
62
|
3expa |
|- ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( ( S i^i dom F ) u. { z } ) ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) |
64 |
63
|
ralrimiva |
|- ( ( ph /\ z e. ( A \ dom F ) ) -> A. w e. ( ( S i^i dom F ) u. { z } ) ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) |
65 |
64
|
adantrr |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> A. w e. ( ( S i^i dom F ) u. { z } ) ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) |
66 |
|
fneq2 |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( C Fn t <-> C Fn ( ( S i^i dom F ) u. { z } ) ) ) |
67 |
|
sseq1 |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( t C_ A <-> ( ( S i^i dom F ) u. { z } ) C_ A ) ) |
68 |
|
sseq2 |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( Pred ( R , A , w ) C_ t <-> Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) ) |
69 |
68
|
raleqbi1dv |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( A. w e. t Pred ( R , A , w ) C_ t <-> A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) ) |
70 |
67 69
|
anbi12d |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) <-> ( ( ( S i^i dom F ) u. { z } ) C_ A /\ A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) ) ) |
71 |
|
raleq |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) <-> A. w e. ( ( S i^i dom F ) u. { z } ) ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) |
72 |
66 70 71
|
3anbi123d |
|- ( t = ( ( S i^i dom F ) u. { z } ) -> ( ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) <-> ( C Fn ( ( S i^i dom F ) u. { z } ) /\ ( ( ( S i^i dom F ) u. { z } ) C_ A /\ A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) /\ A. w e. ( ( S i^i dom F ) u. { z } ) ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) ) |
73 |
72
|
spcegv |
|- ( ( ( S i^i dom F ) u. { z } ) e. _V -> ( ( C Fn ( ( S i^i dom F ) u. { z } ) /\ ( ( ( S i^i dom F ) u. { z } ) C_ A /\ A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) /\ A. w e. ( ( S i^i dom F ) u. { z } ) ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) -> E. t ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) ) |
74 |
73
|
imp |
|- ( ( ( ( S i^i dom F ) u. { z } ) e. _V /\ ( C Fn ( ( S i^i dom F ) u. { z } ) /\ ( ( ( S i^i dom F ) u. { z } ) C_ A /\ A. w e. ( ( S i^i dom F ) u. { z } ) Pred ( R , A , w ) C_ ( ( S i^i dom F ) u. { z } ) ) /\ A. w e. ( ( S i^i dom F ) u. { z } ) ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) -> E. t ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) |
75 |
17 19 61 65 74
|
syl13anc |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> E. t ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) |
76 |
1 2 3
|
frrlem9 |
|- ( ph -> Fun F ) |
77 |
|
resfunexg |
|- ( ( Fun F /\ S e. _V ) -> ( F |` S ) e. _V ) |
78 |
76 12 77
|
syl2an2r |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( F |` S ) e. _V ) |
79 |
|
snex |
|- { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } e. _V |
80 |
|
unexg |
|- ( ( ( F |` S ) e. _V /\ { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } e. _V ) -> ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) e. _V ) |
81 |
78 79 80
|
sylancl |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) e. _V ) |
82 |
4 81
|
eqeltrid |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> C e. _V ) |
83 |
|
fneq1 |
|- ( c = C -> ( c Fn t <-> C Fn t ) ) |
84 |
|
fveq1 |
|- ( c = C -> ( c ` w ) = ( C ` w ) ) |
85 |
|
reseq1 |
|- ( c = C -> ( c |` Pred ( R , A , w ) ) = ( C |` Pred ( R , A , w ) ) ) |
86 |
85
|
oveq2d |
|- ( c = C -> ( w G ( c |` Pred ( R , A , w ) ) ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) |
87 |
84 86
|
eqeq12d |
|- ( c = C -> ( ( c ` w ) = ( w G ( c |` Pred ( R , A , w ) ) ) <-> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) |
88 |
87
|
ralbidv |
|- ( c = C -> ( A. w e. t ( c ` w ) = ( w G ( c |` Pred ( R , A , w ) ) ) <-> A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) |
89 |
83 88
|
3anbi13d |
|- ( c = C -> ( ( c Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( c ` w ) = ( w G ( c |` Pred ( R , A , w ) ) ) ) <-> ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) ) |
90 |
89
|
exbidv |
|- ( c = C -> ( E. t ( c Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( c ` w ) = ( w G ( c |` Pred ( R , A , w ) ) ) ) <-> E. t ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) ) |
91 |
1
|
frrlem1 |
|- B = { c | E. t ( c Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( c ` w ) = ( w G ( c |` Pred ( R , A , w ) ) ) ) } |
92 |
90 91
|
elab2g |
|- ( C e. _V -> ( C e. B <-> E. t ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) ) |
93 |
82 92
|
syl |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> ( C e. B <-> E. t ( C Fn t /\ ( t C_ A /\ A. w e. t Pred ( R , A , w ) C_ t ) /\ A. w e. t ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) ) ) |
94 |
75 93
|
mpbird |
|- ( ( ph /\ ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) -> C e. B ) |