Step |
Hyp |
Ref |
Expression |
1 |
|
ntrcls.o |
|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) ) |
2 |
|
ntrcls.d |
|- D = ( O ` B ) |
3 |
|
ntrcls.r |
|- ( ph -> I D K ) |
4 |
|
sseq1 |
|- ( s = b -> ( s C_ t <-> b C_ t ) ) |
5 |
|
fveq2 |
|- ( s = b -> ( I ` s ) = ( I ` b ) ) |
6 |
5
|
sseq1d |
|- ( s = b -> ( ( I ` s ) C_ ( I ` t ) <-> ( I ` b ) C_ ( I ` t ) ) ) |
7 |
4 6
|
imbi12d |
|- ( s = b -> ( ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> ( b C_ t -> ( I ` b ) C_ ( I ` t ) ) ) ) |
8 |
|
sseq2 |
|- ( t = a -> ( b C_ t <-> b C_ a ) ) |
9 |
|
fveq2 |
|- ( t = a -> ( I ` t ) = ( I ` a ) ) |
10 |
9
|
sseq2d |
|- ( t = a -> ( ( I ` b ) C_ ( I ` t ) <-> ( I ` b ) C_ ( I ` a ) ) ) |
11 |
8 10
|
imbi12d |
|- ( t = a -> ( ( b C_ t -> ( I ` b ) C_ ( I ` t ) ) <-> ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) ) ) |
12 |
7 11
|
cbvral2vw |
|- ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. b e. ~P B A. a e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) ) |
13 |
|
ralcom |
|- ( A. b e. ~P B A. a e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> A. a e. ~P B A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) ) |
14 |
12 13
|
bitri |
|- ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. a e. ~P B A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) ) |
15 |
|
simpl |
|- ( ( ph /\ s e. ~P B ) -> ph ) |
16 |
2 3
|
ntrclsbex |
|- ( ph -> B e. _V ) |
17 |
15 16
|
syl |
|- ( ( ph /\ s e. ~P B ) -> B e. _V ) |
18 |
|
difssd |
|- ( ( ph /\ s e. ~P B ) -> ( B \ s ) C_ B ) |
19 |
17 18
|
sselpwd |
|- ( ( ph /\ s e. ~P B ) -> ( B \ s ) e. ~P B ) |
20 |
|
elpwi |
|- ( a e. ~P B -> a C_ B ) |
21 |
|
simpl |
|- ( ( B e. _V /\ a C_ B ) -> B e. _V ) |
22 |
|
difssd |
|- ( ( B e. _V /\ a C_ B ) -> ( B \ a ) C_ B ) |
23 |
21 22
|
sselpwd |
|- ( ( B e. _V /\ a C_ B ) -> ( B \ a ) e. ~P B ) |
24 |
|
simpr |
|- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> s = ( B \ a ) ) |
25 |
24
|
difeq2d |
|- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> ( B \ s ) = ( B \ ( B \ a ) ) ) |
26 |
25
|
eqeq2d |
|- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> ( a = ( B \ s ) <-> a = ( B \ ( B \ a ) ) ) ) |
27 |
|
eqcom |
|- ( a = ( B \ ( B \ a ) ) <-> ( B \ ( B \ a ) ) = a ) |
28 |
26 27
|
bitrdi |
|- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> ( a = ( B \ s ) <-> ( B \ ( B \ a ) ) = a ) ) |
29 |
|
dfss4 |
|- ( a C_ B <-> ( B \ ( B \ a ) ) = a ) |
30 |
29
|
biimpi |
|- ( a C_ B -> ( B \ ( B \ a ) ) = a ) |
31 |
30
|
adantl |
|- ( ( B e. _V /\ a C_ B ) -> ( B \ ( B \ a ) ) = a ) |
32 |
23 28 31
|
rspcedvd |
|- ( ( B e. _V /\ a C_ B ) -> E. s e. ~P B a = ( B \ s ) ) |
33 |
16 20 32
|
syl2an |
|- ( ( ph /\ a e. ~P B ) -> E. s e. ~P B a = ( B \ s ) ) |
34 |
|
simpl1 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> ph ) |
35 |
34 16
|
syl |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> B e. _V ) |
36 |
|
difssd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> ( B \ t ) C_ B ) |
37 |
35 36
|
sselpwd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> ( B \ t ) e. ~P B ) |
38 |
|
elpwi |
|- ( b e. ~P B -> b C_ B ) |
39 |
|
simpl |
|- ( ( B e. _V /\ b C_ B ) -> B e. _V ) |
40 |
|
difssd |
|- ( ( B e. _V /\ b C_ B ) -> ( B \ b ) C_ B ) |
41 |
39 40
|
sselpwd |
|- ( ( B e. _V /\ b C_ B ) -> ( B \ b ) e. ~P B ) |
42 |
|
simpr |
|- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> t = ( B \ b ) ) |
43 |
42
|
difeq2d |
|- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> ( B \ t ) = ( B \ ( B \ b ) ) ) |
44 |
43
|
eqeq2d |
|- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> ( b = ( B \ t ) <-> b = ( B \ ( B \ b ) ) ) ) |
45 |
|
eqcom |
|- ( b = ( B \ ( B \ b ) ) <-> ( B \ ( B \ b ) ) = b ) |
46 |
44 45
|
bitrdi |
|- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> ( b = ( B \ t ) <-> ( B \ ( B \ b ) ) = b ) ) |
47 |
|
dfss4 |
|- ( b C_ B <-> ( B \ ( B \ b ) ) = b ) |
48 |
47
|
biimpi |
|- ( b C_ B -> ( B \ ( B \ b ) ) = b ) |
49 |
48
|
adantl |
|- ( ( B e. _V /\ b C_ B ) -> ( B \ ( B \ b ) ) = b ) |
50 |
41 46 49
|
rspcedvd |
|- ( ( B e. _V /\ b C_ B ) -> E. t e. ~P B b = ( B \ t ) ) |
51 |
16 38 50
|
syl2an |
|- ( ( ph /\ b e. ~P B ) -> E. t e. ~P B b = ( B \ t ) ) |
52 |
51
|
3ad2antl1 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ b e. ~P B ) -> E. t e. ~P B b = ( B \ t ) ) |
53 |
|
simp12 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> s e. ~P B ) |
54 |
53
|
elpwid |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> s C_ B ) |
55 |
|
simp2 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> t e. ~P B ) |
56 |
55
|
elpwid |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> t C_ B ) |
57 |
|
sscon34b |
|- ( ( s C_ B /\ t C_ B ) -> ( s C_ t <-> ( B \ t ) C_ ( B \ s ) ) ) |
58 |
54 56 57
|
syl2anc |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( s C_ t <-> ( B \ t ) C_ ( B \ s ) ) ) |
59 |
58
|
bicomd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( B \ t ) C_ ( B \ s ) <-> s C_ t ) ) |
60 |
|
simp11 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ph ) |
61 |
1 2 3
|
ntrclsiex |
|- ( ph -> I e. ( ~P B ^m ~P B ) ) |
62 |
60 61
|
syl |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> I e. ( ~P B ^m ~P B ) ) |
63 |
|
elmapi |
|- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B ) |
64 |
62 63
|
syl |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> I : ~P B --> ~P B ) |
65 |
60 16
|
syl |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> B e. _V ) |
66 |
|
difssd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ t ) C_ B ) |
67 |
65 66
|
sselpwd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ t ) e. ~P B ) |
68 |
64 67
|
ffvelrnd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ t ) ) e. ~P B ) |
69 |
68
|
elpwid |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ t ) ) C_ B ) |
70 |
|
difssd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ s ) C_ B ) |
71 |
65 70
|
sselpwd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ s ) e. ~P B ) |
72 |
64 71
|
ffvelrnd |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ s ) ) e. ~P B ) |
73 |
72
|
elpwid |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ s ) ) C_ B ) |
74 |
|
sscon34b |
|- ( ( ( I ` ( B \ t ) ) C_ B /\ ( I ` ( B \ s ) ) C_ B ) -> ( ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) <-> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) |
75 |
69 73 74
|
syl2anc |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) <-> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) |
76 |
59 75
|
imbi12d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( ( B \ t ) C_ ( B \ s ) -> ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) ) <-> ( s C_ t -> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) ) |
77 |
|
simp3 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> b = ( B \ t ) ) |
78 |
|
simp13 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> a = ( B \ s ) ) |
79 |
77 78
|
sseq12d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( b C_ a <-> ( B \ t ) C_ ( B \ s ) ) ) |
80 |
77
|
fveq2d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` b ) = ( I ` ( B \ t ) ) ) |
81 |
78
|
fveq2d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` a ) = ( I ` ( B \ s ) ) ) |
82 |
80 81
|
sseq12d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` b ) C_ ( I ` a ) <-> ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) ) ) |
83 |
79 82
|
imbi12d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> ( ( B \ t ) C_ ( B \ s ) -> ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) ) ) ) |
84 |
1 2 3
|
ntrclsfv1 |
|- ( ph -> ( D ` I ) = K ) |
85 |
60 84
|
syl |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( D ` I ) = K ) |
86 |
85
|
fveq1d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` s ) = ( K ` s ) ) |
87 |
|
eqid |
|- ( D ` I ) = ( D ` I ) |
88 |
|
eqid |
|- ( ( D ` I ) ` s ) = ( ( D ` I ) ` s ) |
89 |
1 2 65 62 87 53 88
|
dssmapfv3d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` s ) = ( B \ ( I ` ( B \ s ) ) ) ) |
90 |
86 89
|
eqtr3d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( K ` s ) = ( B \ ( I ` ( B \ s ) ) ) ) |
91 |
60 3
|
syl |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> I D K ) |
92 |
1 2 91
|
ntrclsfv1 |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( D ` I ) = K ) |
93 |
92
|
fveq1d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` t ) = ( K ` t ) ) |
94 |
|
eqid |
|- ( ( D ` I ) ` t ) = ( ( D ` I ) ` t ) |
95 |
1 2 65 62 87 55 94
|
dssmapfv3d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` t ) = ( B \ ( I ` ( B \ t ) ) ) ) |
96 |
93 95
|
eqtr3d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( K ` t ) = ( B \ ( I ` ( B \ t ) ) ) ) |
97 |
90 96
|
sseq12d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( K ` s ) C_ ( K ` t ) <-> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) |
98 |
97
|
imbi2d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) <-> ( s C_ t -> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) ) |
99 |
76 83 98
|
3bitr4d |
|- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) ) |
100 |
37 52 99
|
ralxfrd2 |
|- ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) -> ( A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) ) |
101 |
19 33 100
|
ralxfrd2 |
|- ( ph -> ( A. a e. ~P B A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> A. s e. ~P B A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) ) |
102 |
14 101
|
syl5bb |
|- ( ph -> ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) ) |