Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1145.1 |
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) ) |
2 |
|
bnj1145.2 |
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) |
3 |
|
bnj1145.3 |
|- D = ( _om \ { (/) } ) |
4 |
|
bnj1145.4 |
|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) } |
5 |
|
bnj1145.5 |
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) ) |
6 |
|
bnj1145.6 |
|- ( th <-> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) |
7 |
1 2 3 4
|
bnj882 |
|- _trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i ) |
8 |
|
ss2iun |
|- ( A. f e. B U_ i e. dom f ( f ` i ) C_ A -> U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A ) |
9 |
5 4
|
bnj1083 |
|- ( f e. B <-> E. n ch ) |
10 |
2
|
bnj1095 |
|- ( ps -> A. i ps ) |
11 |
10 5
|
bnj1096 |
|- ( ch -> A. i ch ) |
12 |
3
|
bnj1098 |
|- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) ) |
13 |
5
|
bnj1232 |
|- ( ch -> n e. D ) |
14 |
13
|
3anim3i |
|- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( i =/= (/) /\ i e. n /\ n e. D ) ) |
15 |
12 14
|
bnj1101 |
|- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) ) |
16 |
|
ancl |
|- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) ) -> ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) ) |
17 |
15 16
|
bnj101 |
|- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) |
18 |
6
|
imbi2i |
|- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> th ) <-> ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) ) |
19 |
18
|
exbii |
|- ( E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> th ) <-> E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) ) |
20 |
17 19
|
mpbir |
|- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> th ) |
21 |
|
bnj213 |
|- _pred ( y , A , R ) C_ A |
22 |
21
|
bnj226 |
|- U_ y e. ( f ` j ) _pred ( y , A , R ) C_ A |
23 |
|
simpr |
|- ( ( j e. n /\ i = suc j ) -> i = suc j ) |
24 |
6 23
|
simplbiim |
|- ( th -> i = suc j ) |
25 |
|
simp2 |
|- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. n ) |
26 |
13
|
3ad2ant3 |
|- ( ( i =/= (/) /\ i e. n /\ ch ) -> n e. D ) |
27 |
3
|
bnj923 |
|- ( n e. D -> n e. _om ) |
28 |
|
elnn |
|- ( ( i e. n /\ n e. _om ) -> i e. _om ) |
29 |
27 28
|
sylan2 |
|- ( ( i e. n /\ n e. D ) -> i e. _om ) |
30 |
25 26 29
|
syl2anc |
|- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. _om ) |
31 |
6 30
|
bnj832 |
|- ( th -> i e. _om ) |
32 |
|
vex |
|- j e. _V |
33 |
32
|
bnj216 |
|- ( i = suc j -> j e. i ) |
34 |
|
elnn |
|- ( ( j e. i /\ i e. _om ) -> j e. _om ) |
35 |
33 34
|
sylan |
|- ( ( i = suc j /\ i e. _om ) -> j e. _om ) |
36 |
24 31 35
|
syl2anc |
|- ( th -> j e. _om ) |
37 |
6 25
|
bnj832 |
|- ( th -> i e. n ) |
38 |
24 37
|
eqeltrrd |
|- ( th -> suc j e. n ) |
39 |
2
|
bnj589 |
|- ( ps <-> A. j e. _om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) |
40 |
39
|
biimpi |
|- ( ps -> A. j e. _om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) |
41 |
40
|
bnj708 |
|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> A. j e. _om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) |
42 |
|
rsp |
|- ( A. j e. _om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) -> ( j e. _om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) ) |
43 |
41 42
|
syl |
|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( j e. _om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) ) |
44 |
5 43
|
sylbi |
|- ( ch -> ( j e. _om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) ) |
45 |
44
|
3ad2ant3 |
|- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. _om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) ) |
46 |
6 45
|
bnj832 |
|- ( th -> ( j e. _om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) ) |
47 |
36 38 46
|
mp2d |
|- ( th -> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) |
48 |
|
fveqeq2 |
|- ( i = suc j -> ( ( f ` i ) = U_ y e. ( f ` j ) _pred ( y , A , R ) <-> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) |
49 |
24 48
|
syl |
|- ( th -> ( ( f ` i ) = U_ y e. ( f ` j ) _pred ( y , A , R ) <-> ( f ` suc j ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) ) |
50 |
47 49
|
mpbird |
|- ( th -> ( f ` i ) = U_ y e. ( f ` j ) _pred ( y , A , R ) ) |
51 |
22 50
|
bnj1262 |
|- ( th -> ( f ` i ) C_ A ) |
52 |
20 51
|
bnj1023 |
|- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) |
53 |
|
3anass |
|- ( ( i =/= (/) /\ i e. n /\ ch ) <-> ( i =/= (/) /\ ( i e. n /\ ch ) ) ) |
54 |
53
|
imbi1i |
|- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) ) |
55 |
54
|
exbii |
|- ( E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) ) |
56 |
52 55
|
mpbi |
|- E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) |
57 |
1
|
biimpi |
|- ( ph -> ( f ` (/) ) = _pred ( X , A , R ) ) |
58 |
5 57
|
bnj771 |
|- ( ch -> ( f ` (/) ) = _pred ( X , A , R ) ) |
59 |
|
fveq2 |
|- ( i = (/) -> ( f ` i ) = ( f ` (/) ) ) |
60 |
|
bnj213 |
|- _pred ( X , A , R ) C_ A |
61 |
|
sseq1 |
|- ( ( f ` (/) ) = _pred ( X , A , R ) -> ( ( f ` (/) ) C_ A <-> _pred ( X , A , R ) C_ A ) ) |
62 |
60 61
|
mpbiri |
|- ( ( f ` (/) ) = _pred ( X , A , R ) -> ( f ` (/) ) C_ A ) |
63 |
|
sseq1 |
|- ( ( f ` i ) = ( f ` (/) ) -> ( ( f ` i ) C_ A <-> ( f ` (/) ) C_ A ) ) |
64 |
63
|
biimpar |
|- ( ( ( f ` i ) = ( f ` (/) ) /\ ( f ` (/) ) C_ A ) -> ( f ` i ) C_ A ) |
65 |
59 62 64
|
syl2an |
|- ( ( i = (/) /\ ( f ` (/) ) = _pred ( X , A , R ) ) -> ( f ` i ) C_ A ) |
66 |
58 65
|
sylan2 |
|- ( ( i = (/) /\ ch ) -> ( f ` i ) C_ A ) |
67 |
66
|
adantrl |
|- ( ( i = (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) |
68 |
56 67
|
bnj1109 |
|- E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) |
69 |
|
19.9v |
|- ( E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) ) |
70 |
68 69
|
mpbi |
|- ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) |
71 |
70
|
expcom |
|- ( ch -> ( i e. n -> ( f ` i ) C_ A ) ) |
72 |
|
fndm |
|- ( f Fn n -> dom f = n ) |
73 |
5 72
|
bnj770 |
|- ( ch -> dom f = n ) |
74 |
|
eleq2 |
|- ( dom f = n -> ( i e. dom f <-> i e. n ) ) |
75 |
74
|
imbi1d |
|- ( dom f = n -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) ) |
76 |
73 75
|
syl |
|- ( ch -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) ) |
77 |
71 76
|
mpbird |
|- ( ch -> ( i e. dom f -> ( f ` i ) C_ A ) ) |
78 |
11 77
|
hbralrimi |
|- ( ch -> A. i e. dom f ( f ` i ) C_ A ) |
79 |
78
|
exlimiv |
|- ( E. n ch -> A. i e. dom f ( f ` i ) C_ A ) |
80 |
9 79
|
sylbi |
|- ( f e. B -> A. i e. dom f ( f ` i ) C_ A ) |
81 |
|
ss2iun |
|- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ U_ i e. dom f A ) |
82 |
|
bnj1143 |
|- U_ i e. dom f A C_ A |
83 |
81 82
|
sstrdi |
|- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ A ) |
84 |
80 83
|
syl |
|- ( f e. B -> U_ i e. dom f ( f ` i ) C_ A ) |
85 |
8 84
|
mprg |
|- U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A |
86 |
4
|
bnj1317 |
|- ( w e. B -> A. f w e. B ) |
87 |
86
|
bnj1146 |
|- U_ f e. B A C_ A |
88 |
85 87
|
sstri |
|- U_ f e. B U_ i e. dom f ( f ` i ) C_ A |
89 |
7 88
|
eqsstri |
|- _trCl ( X , A , R ) C_ A |