Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1408.1 |
|- B = ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
2 |
|
bnj1408.2 |
|- C = ( _pred ( X , A , R ) u. U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) ) |
3 |
|
bnj1408.3 |
|- ( th <-> ( R _FrSe A /\ X e. A ) ) |
4 |
|
bnj1408.4 |
|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) ) |
5 |
3
|
biimpri |
|- ( ( R _FrSe A /\ X e. A ) -> th ) |
6 |
1
|
bnj1413 |
|- ( ( R _FrSe A /\ X e. A ) -> B e. _V ) |
7 |
|
simplll |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. _pred ( X , A , R ) ) -> R _FrSe A ) |
8 |
|
bnj213 |
|- _pred ( X , A , R ) C_ A |
9 |
8
|
sseli |
|- ( z e. _pred ( X , A , R ) -> z e. A ) |
10 |
9
|
adantl |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. _pred ( X , A , R ) ) -> z e. A ) |
11 |
|
bnj906 |
|- ( ( R _FrSe A /\ z e. A ) -> _pred ( z , A , R ) C_ _trCl ( z , A , R ) ) |
12 |
7 10 11
|
syl2anc |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. _pred ( X , A , R ) ) -> _pred ( z , A , R ) C_ _trCl ( z , A , R ) ) |
13 |
|
bnj1318 |
|- ( y = z -> _trCl ( y , A , R ) = _trCl ( z , A , R ) ) |
14 |
13
|
ssiun2s |
|- ( z e. _pred ( X , A , R ) -> _trCl ( z , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
15 |
|
ssun4 |
|- ( _trCl ( z , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) -> _trCl ( z , A , R ) C_ ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) ) |
16 |
15 1
|
sseqtrrdi |
|- ( _trCl ( z , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) -> _trCl ( z , A , R ) C_ B ) |
17 |
14 16
|
syl |
|- ( z e. _pred ( X , A , R ) -> _trCl ( z , A , R ) C_ B ) |
18 |
17
|
adantl |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. _pred ( X , A , R ) ) -> _trCl ( z , A , R ) C_ B ) |
19 |
12 18
|
sstrd |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. _pred ( X , A , R ) ) -> _pred ( z , A , R ) C_ B ) |
20 |
|
simpr |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
21 |
20
|
bnj1405 |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> E. y e. _pred ( X , A , R ) z e. _trCl ( y , A , R ) ) |
22 |
|
biid |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) <-> ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) ) |
23 |
|
nfv |
|- F/ y ( R _FrSe A /\ X e. A ) |
24 |
|
nfcv |
|- F/_ y _pred ( X , A , R ) |
25 |
|
nfiu1 |
|- F/_ y U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) |
26 |
24 25
|
nfun |
|- F/_ y ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
27 |
1 26
|
nfcxfr |
|- F/_ y B |
28 |
27
|
nfcri |
|- F/ y z e. B |
29 |
23 28
|
nfan |
|- F/ y ( ( R _FrSe A /\ X e. A ) /\ z e. B ) |
30 |
25
|
nfcri |
|- F/ y z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) |
31 |
29 30
|
nfan |
|- F/ y ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
32 |
31
|
nf5ri |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> A. y ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) ) |
33 |
21 22 32
|
bnj1521 |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> E. y ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) ) |
34 |
|
simplll |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> R _FrSe A ) |
35 |
34
|
3ad2ant1 |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> R _FrSe A ) |
36 |
|
bnj1147 |
|- _trCl ( y , A , R ) C_ A |
37 |
|
simp3 |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> z e. _trCl ( y , A , R ) ) |
38 |
36 37
|
bnj1213 |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> z e. A ) |
39 |
35 38 11
|
syl2anc |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> _pred ( z , A , R ) C_ _trCl ( z , A , R ) ) |
40 |
|
simp2 |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> y e. _pred ( X , A , R ) ) |
41 |
8 40
|
bnj1213 |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> y e. A ) |
42 |
|
bnj1125 |
|- ( ( R _FrSe A /\ y e. A /\ z e. _trCl ( y , A , R ) ) -> _trCl ( z , A , R ) C_ _trCl ( y , A , R ) ) |
43 |
35 41 37 42
|
syl3anc |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> _trCl ( z , A , R ) C_ _trCl ( y , A , R ) ) |
44 |
39 43
|
sstrd |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> _pred ( z , A , R ) C_ _trCl ( y , A , R ) ) |
45 |
|
ssiun2 |
|- ( y e. _pred ( X , A , R ) -> _trCl ( y , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
46 |
45
|
3ad2ant2 |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> _trCl ( y , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) |
47 |
|
ssun4 |
|- ( _trCl ( y , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) -> _trCl ( y , A , R ) C_ ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) ) |
48 |
47 1
|
sseqtrrdi |
|- ( _trCl ( y , A , R ) C_ U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) -> _trCl ( y , A , R ) C_ B ) |
49 |
46 48
|
syl |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> _trCl ( y , A , R ) C_ B ) |
50 |
44 49
|
sstrd |
|- ( ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) /\ y e. _pred ( X , A , R ) /\ z e. _trCl ( y , A , R ) ) -> _pred ( z , A , R ) C_ B ) |
51 |
33 50
|
bnj593 |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> E. y _pred ( z , A , R ) C_ B ) |
52 |
|
nfcv |
|- F/_ y _pred ( z , A , R ) |
53 |
52 27
|
nfss |
|- F/ y _pred ( z , A , R ) C_ B |
54 |
53
|
nf5ri |
|- ( _pred ( z , A , R ) C_ B -> A. y _pred ( z , A , R ) C_ B ) |
55 |
51 54
|
bnj1397 |
|- ( ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) -> _pred ( z , A , R ) C_ B ) |
56 |
|
simpr |
|- ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) -> z e. B ) |
57 |
1
|
bnj1138 |
|- ( z e. B <-> ( z e. _pred ( X , A , R ) \/ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) ) |
58 |
56 57
|
sylib |
|- ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) -> ( z e. _pred ( X , A , R ) \/ z e. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) ) |
59 |
19 55 58
|
mpjaodan |
|- ( ( ( R _FrSe A /\ X e. A ) /\ z e. B ) -> _pred ( z , A , R ) C_ B ) |
60 |
59
|
ralrimiva |
|- ( ( R _FrSe A /\ X e. A ) -> A. z e. B _pred ( z , A , R ) C_ B ) |
61 |
|
df-bnj19 |
|- ( _TrFo ( B , A , R ) <-> A. z e. B _pred ( z , A , R ) C_ B ) |
62 |
60 61
|
sylibr |
|- ( ( R _FrSe A /\ X e. A ) -> _TrFo ( B , A , R ) ) |
63 |
1
|
bnj931 |
|- _pred ( X , A , R ) C_ B |
64 |
63
|
a1i |
|- ( ( R _FrSe A /\ X e. A ) -> _pred ( X , A , R ) C_ B ) |
65 |
6 62 64 4
|
syl3anbrc |
|- ( ( R _FrSe A /\ X e. A ) -> ta ) |
66 |
3 4
|
bnj1124 |
|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B ) |
67 |
5 65 66
|
syl2anc |
|- ( ( R _FrSe A /\ X e. A ) -> _trCl ( X , A , R ) C_ B ) |
68 |
|
bnj906 |
|- ( ( R _FrSe A /\ X e. A ) -> _pred ( X , A , R ) C_ _trCl ( X , A , R ) ) |
69 |
|
iunss1 |
|- ( _pred ( X , A , R ) C_ _trCl ( X , A , R ) -> U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) C_ U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) ) |
70 |
|
unss2 |
|- ( U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) C_ U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) -> ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) C_ ( _pred ( X , A , R ) u. U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) ) ) |
71 |
68 69 70
|
3syl |
|- ( ( R _FrSe A /\ X e. A ) -> ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) ) C_ ( _pred ( X , A , R ) u. U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) ) ) |
72 |
71 1 2
|
3sstr4g |
|- ( ( R _FrSe A /\ X e. A ) -> B C_ C ) |
73 |
|
biid |
|- ( ( R _FrSe A /\ X e. A ) <-> ( R _FrSe A /\ X e. A ) ) |
74 |
|
biid |
|- ( ( C e. _V /\ _TrFo ( C , A , R ) /\ _pred ( X , A , R ) C_ C ) <-> ( C e. _V /\ _TrFo ( C , A , R ) /\ _pred ( X , A , R ) C_ C ) ) |
75 |
2 73 74
|
bnj1136 |
|- ( ( R _FrSe A /\ X e. A ) -> _trCl ( X , A , R ) = C ) |
76 |
72 75
|
sseqtrrd |
|- ( ( R _FrSe A /\ X e. A ) -> B C_ _trCl ( X , A , R ) ) |
77 |
67 76
|
eqssd |
|- ( ( R _FrSe A /\ X e. A ) -> _trCl ( X , A , R ) = B ) |