Step |
Hyp |
Ref |
Expression |
1 |
|
bnj557.3 |
|- D = ( _om \ { (/) } ) |
2 |
|
bnj557.16 |
|- G = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) |
3 |
|
bnj557.17 |
|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) |
4 |
|
bnj557.18 |
|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) ) |
5 |
|
bnj557.19 |
|- ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) ) |
6 |
|
bnj557.20 |
|- ( ze <-> ( i e. _om /\ suc i e. n /\ m = suc i ) ) |
7 |
|
bnj557.21 |
|- B = U_ y e. ( f ` i ) _pred ( y , A , R ) |
8 |
|
bnj557.22 |
|- C = U_ y e. ( f ` p ) _pred ( y , A , R ) |
9 |
|
bnj557.23 |
|- K = U_ y e. ( G ` i ) _pred ( y , A , R ) |
10 |
|
bnj557.24 |
|- L = U_ y e. ( G ` p ) _pred ( y , A , R ) |
11 |
|
bnj557.25 |
|- G = ( f u. { <. m , C >. } ) |
12 |
|
bnj557.28 |
|- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) ) |
13 |
|
bnj557.29 |
|- ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) |
14 |
|
bnj557.36 |
|- ( ( R _FrSe A /\ ta /\ si ) -> G Fn n ) |
15 |
|
3an4anass |
|- ( ( ( R _FrSe A /\ ta /\ et ) /\ ze ) <-> ( ( R _FrSe A /\ ta ) /\ ( et /\ ze ) ) ) |
16 |
4 5
|
bnj556 |
|- ( et -> si ) |
17 |
16
|
3anim3i |
|- ( ( R _FrSe A /\ ta /\ et ) -> ( R _FrSe A /\ ta /\ si ) ) |
18 |
|
vex |
|- i e. _V |
19 |
18
|
bnj216 |
|- ( m = suc i -> i e. m ) |
20 |
6 19
|
bnj837 |
|- ( ze -> i e. m ) |
21 |
17 20
|
anim12i |
|- ( ( ( R _FrSe A /\ ta /\ et ) /\ ze ) -> ( ( R _FrSe A /\ ta /\ si ) /\ i e. m ) ) |
22 |
15 21
|
sylbir |
|- ( ( ( R _FrSe A /\ ta ) /\ ( et /\ ze ) ) -> ( ( R _FrSe A /\ ta /\ si ) /\ i e. m ) ) |
23 |
5
|
bnj1254 |
|- ( et -> m = suc p ) |
24 |
6
|
simp3bi |
|- ( ze -> m = suc i ) |
25 |
|
bnj551 |
|- ( ( m = suc p /\ m = suc i ) -> p = i ) |
26 |
23 24 25
|
syl2an |
|- ( ( et /\ ze ) -> p = i ) |
27 |
26
|
adantl |
|- ( ( ( R _FrSe A /\ ta ) /\ ( et /\ ze ) ) -> p = i ) |
28 |
22 27
|
jca |
|- ( ( ( R _FrSe A /\ ta ) /\ ( et /\ ze ) ) -> ( ( ( R _FrSe A /\ ta /\ si ) /\ i e. m ) /\ p = i ) ) |
29 |
|
bnj256 |
|- ( ( R _FrSe A /\ ta /\ et /\ ze ) <-> ( ( R _FrSe A /\ ta ) /\ ( et /\ ze ) ) ) |
30 |
|
df-3an |
|- ( ( ( R _FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) <-> ( ( ( R _FrSe A /\ ta /\ si ) /\ i e. m ) /\ p = i ) ) |
31 |
28 29 30
|
3imtr4i |
|- ( ( R _FrSe A /\ ta /\ et /\ ze ) -> ( ( R _FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) ) |
32 |
12 13 1 2 3 4 8 11 7 9 10 14
|
bnj553 |
|- ( ( ( R _FrSe A /\ ta /\ si ) /\ i e. m /\ p = i ) -> ( G ` m ) = L ) |
33 |
31 32
|
syl |
|- ( ( R _FrSe A /\ ta /\ et /\ ze ) -> ( G ` m ) = L ) |