Step |
Hyp |
Ref |
Expression |
1 |
|
inss2 |
|- ( .<_ i^i `' .<_ ) C_ `' .<_ |
2 |
|
relcnv |
|- Rel `' .<_ |
3 |
|
relss |
|- ( ( .<_ i^i `' .<_ ) C_ `' .<_ -> ( Rel `' .<_ -> Rel ( .<_ i^i `' .<_ ) ) ) |
4 |
1 2 3
|
mp2 |
|- Rel ( .<_ i^i `' .<_ ) |
5 |
4
|
a1i |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> Rel ( .<_ i^i `' .<_ ) ) |
6 |
|
eqidd |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> dom ( .<_ i^i `' .<_ ) = dom ( .<_ i^i `' .<_ ) ) |
7 |
|
brin |
|- ( r ( .<_ i^i `' .<_ ) s <-> ( r .<_ s /\ r `' .<_ s ) ) |
8 |
|
vex |
|- r e. _V |
9 |
|
vex |
|- s e. _V |
10 |
8 9
|
brcnv |
|- ( r `' .<_ s <-> s .<_ r ) |
11 |
10
|
anbi2i |
|- ( ( r .<_ s /\ r `' .<_ s ) <-> ( r .<_ s /\ s .<_ r ) ) |
12 |
7 11
|
bitri |
|- ( r ( .<_ i^i `' .<_ ) s <-> ( r .<_ s /\ s .<_ r ) ) |
13 |
|
brin |
|- ( s ( .<_ i^i `' .<_ ) t <-> ( s .<_ t /\ s `' .<_ t ) ) |
14 |
|
vex |
|- t e. _V |
15 |
9 14
|
brcnv |
|- ( s `' .<_ t <-> t .<_ s ) |
16 |
15
|
anbi2i |
|- ( ( s .<_ t /\ s `' .<_ t ) <-> ( s .<_ t /\ t .<_ s ) ) |
17 |
13 16
|
bitri |
|- ( s ( .<_ i^i `' .<_ ) t <-> ( s .<_ t /\ t .<_ s ) ) |
18 |
12 17
|
anbi12i |
|- ( ( r ( .<_ i^i `' .<_ ) s /\ s ( .<_ i^i `' .<_ ) t ) <-> ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) ) |
19 |
|
breq1 |
|- ( a = r -> ( a .<_ b <-> r .<_ b ) ) |
20 |
19
|
anbi1d |
|- ( a = r -> ( ( a .<_ b /\ b .<_ c ) <-> ( r .<_ b /\ b .<_ c ) ) ) |
21 |
|
breq1 |
|- ( a = r -> ( a .<_ c <-> r .<_ c ) ) |
22 |
20 21
|
imbi12d |
|- ( a = r -> ( ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) <-> ( ( r .<_ b /\ b .<_ c ) -> r .<_ c ) ) ) |
23 |
22
|
2albidv |
|- ( a = r -> ( A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) <-> A. b A. c ( ( r .<_ b /\ b .<_ c ) -> r .<_ c ) ) ) |
24 |
23
|
spvv |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> A. b A. c ( ( r .<_ b /\ b .<_ c ) -> r .<_ c ) ) |
25 |
|
breq2 |
|- ( b = s -> ( r .<_ b <-> r .<_ s ) ) |
26 |
|
breq1 |
|- ( b = s -> ( b .<_ c <-> s .<_ c ) ) |
27 |
25 26
|
anbi12d |
|- ( b = s -> ( ( r .<_ b /\ b .<_ c ) <-> ( r .<_ s /\ s .<_ c ) ) ) |
28 |
27
|
imbi1d |
|- ( b = s -> ( ( ( r .<_ b /\ b .<_ c ) -> r .<_ c ) <-> ( ( r .<_ s /\ s .<_ c ) -> r .<_ c ) ) ) |
29 |
28
|
albidv |
|- ( b = s -> ( A. c ( ( r .<_ b /\ b .<_ c ) -> r .<_ c ) <-> A. c ( ( r .<_ s /\ s .<_ c ) -> r .<_ c ) ) ) |
30 |
29
|
spvv |
|- ( A. b A. c ( ( r .<_ b /\ b .<_ c ) -> r .<_ c ) -> A. c ( ( r .<_ s /\ s .<_ c ) -> r .<_ c ) ) |
31 |
|
breq2 |
|- ( c = t -> ( s .<_ c <-> s .<_ t ) ) |
32 |
31
|
anbi2d |
|- ( c = t -> ( ( r .<_ s /\ s .<_ c ) <-> ( r .<_ s /\ s .<_ t ) ) ) |
33 |
|
breq2 |
|- ( c = t -> ( r .<_ c <-> r .<_ t ) ) |
34 |
32 33
|
imbi12d |
|- ( c = t -> ( ( ( r .<_ s /\ s .<_ c ) -> r .<_ c ) <-> ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) ) ) |
35 |
34
|
spvv |
|- ( A. c ( ( r .<_ s /\ s .<_ c ) -> r .<_ c ) -> ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) ) |
36 |
|
pm3.3 |
|- ( ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) -> ( r .<_ s -> ( s .<_ t -> r .<_ t ) ) ) |
37 |
36
|
com23 |
|- ( ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) -> ( s .<_ t -> ( r .<_ s -> r .<_ t ) ) ) |
38 |
37
|
adantrd |
|- ( ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) -> ( ( s .<_ t /\ t .<_ s ) -> ( r .<_ s -> r .<_ t ) ) ) |
39 |
38
|
com23 |
|- ( ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) -> ( r .<_ s -> ( ( s .<_ t /\ t .<_ s ) -> r .<_ t ) ) ) |
40 |
39
|
adantrd |
|- ( ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) -> ( ( r .<_ s /\ s .<_ r ) -> ( ( s .<_ t /\ t .<_ s ) -> r .<_ t ) ) ) |
41 |
40
|
impd |
|- ( ( ( r .<_ s /\ s .<_ t ) -> r .<_ t ) -> ( ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) -> r .<_ t ) ) |
42 |
24 30 35 41
|
4syl |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) -> r .<_ t ) ) |
43 |
|
breq1 |
|- ( a = t -> ( a .<_ b <-> t .<_ b ) ) |
44 |
43
|
anbi1d |
|- ( a = t -> ( ( a .<_ b /\ b .<_ c ) <-> ( t .<_ b /\ b .<_ c ) ) ) |
45 |
|
breq1 |
|- ( a = t -> ( a .<_ c <-> t .<_ c ) ) |
46 |
44 45
|
imbi12d |
|- ( a = t -> ( ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) <-> ( ( t .<_ b /\ b .<_ c ) -> t .<_ c ) ) ) |
47 |
46
|
2albidv |
|- ( a = t -> ( A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) <-> A. b A. c ( ( t .<_ b /\ b .<_ c ) -> t .<_ c ) ) ) |
48 |
47
|
spvv |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> A. b A. c ( ( t .<_ b /\ b .<_ c ) -> t .<_ c ) ) |
49 |
|
breq2 |
|- ( b = s -> ( t .<_ b <-> t .<_ s ) ) |
50 |
49 26
|
anbi12d |
|- ( b = s -> ( ( t .<_ b /\ b .<_ c ) <-> ( t .<_ s /\ s .<_ c ) ) ) |
51 |
50
|
imbi1d |
|- ( b = s -> ( ( ( t .<_ b /\ b .<_ c ) -> t .<_ c ) <-> ( ( t .<_ s /\ s .<_ c ) -> t .<_ c ) ) ) |
52 |
51
|
albidv |
|- ( b = s -> ( A. c ( ( t .<_ b /\ b .<_ c ) -> t .<_ c ) <-> A. c ( ( t .<_ s /\ s .<_ c ) -> t .<_ c ) ) ) |
53 |
52
|
spvv |
|- ( A. b A. c ( ( t .<_ b /\ b .<_ c ) -> t .<_ c ) -> A. c ( ( t .<_ s /\ s .<_ c ) -> t .<_ c ) ) |
54 |
|
breq2 |
|- ( c = r -> ( s .<_ c <-> s .<_ r ) ) |
55 |
54
|
anbi2d |
|- ( c = r -> ( ( t .<_ s /\ s .<_ c ) <-> ( t .<_ s /\ s .<_ r ) ) ) |
56 |
|
breq2 |
|- ( c = r -> ( t .<_ c <-> t .<_ r ) ) |
57 |
55 56
|
imbi12d |
|- ( c = r -> ( ( ( t .<_ s /\ s .<_ c ) -> t .<_ c ) <-> ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) ) ) |
58 |
57
|
spvv |
|- ( A. c ( ( t .<_ s /\ s .<_ c ) -> t .<_ c ) -> ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) ) |
59 |
|
pm3.3 |
|- ( ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) -> ( t .<_ s -> ( s .<_ r -> t .<_ r ) ) ) |
60 |
59
|
adantld |
|- ( ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) -> ( ( s .<_ t /\ t .<_ s ) -> ( s .<_ r -> t .<_ r ) ) ) |
61 |
60
|
com23 |
|- ( ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) -> ( s .<_ r -> ( ( s .<_ t /\ t .<_ s ) -> t .<_ r ) ) ) |
62 |
61
|
adantld |
|- ( ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) -> ( ( r .<_ s /\ s .<_ r ) -> ( ( s .<_ t /\ t .<_ s ) -> t .<_ r ) ) ) |
63 |
62
|
impd |
|- ( ( ( t .<_ s /\ s .<_ r ) -> t .<_ r ) -> ( ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) -> t .<_ r ) ) |
64 |
48 53 58 63
|
4syl |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) -> t .<_ r ) ) |
65 |
42 64
|
jcad |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) -> ( r .<_ t /\ t .<_ r ) ) ) |
66 |
|
brin |
|- ( r ( .<_ i^i `' .<_ ) t <-> ( r .<_ t /\ r `' .<_ t ) ) |
67 |
8 14
|
brcnv |
|- ( r `' .<_ t <-> t .<_ r ) |
68 |
67
|
anbi2i |
|- ( ( r .<_ t /\ r `' .<_ t ) <-> ( r .<_ t /\ t .<_ r ) ) |
69 |
66 68
|
bitr2i |
|- ( ( r .<_ t /\ t .<_ r ) <-> r ( .<_ i^i `' .<_ ) t ) |
70 |
65 69
|
syl6ib |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( ( ( r .<_ s /\ s .<_ r ) /\ ( s .<_ t /\ t .<_ s ) ) -> r ( .<_ i^i `' .<_ ) t ) ) |
71 |
18 70
|
syl5bi |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( ( r ( .<_ i^i `' .<_ ) s /\ s ( .<_ i^i `' .<_ ) t ) -> r ( .<_ i^i `' .<_ ) t ) ) |
72 |
9 8
|
brcnv |
|- ( s `' .<_ r <-> r .<_ s ) |
73 |
72
|
bicomi |
|- ( r .<_ s <-> s `' .<_ r ) |
74 |
73 10
|
anbi12ci |
|- ( ( r .<_ s /\ r `' .<_ s ) <-> ( s .<_ r /\ s `' .<_ r ) ) |
75 |
|
brin |
|- ( s ( .<_ i^i `' .<_ ) r <-> ( s .<_ r /\ s `' .<_ r ) ) |
76 |
74 7 75
|
3bitr4i |
|- ( r ( .<_ i^i `' .<_ ) s <-> s ( .<_ i^i `' .<_ ) r ) |
77 |
76
|
biimpi |
|- ( r ( .<_ i^i `' .<_ ) s -> s ( .<_ i^i `' .<_ ) r ) |
78 |
71 77
|
jctil |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( ( r ( .<_ i^i `' .<_ ) s -> s ( .<_ i^i `' .<_ ) r ) /\ ( ( r ( .<_ i^i `' .<_ ) s /\ s ( .<_ i^i `' .<_ ) t ) -> r ( .<_ i^i `' .<_ ) t ) ) ) |
79 |
78
|
alrimiv |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> A. t ( ( r ( .<_ i^i `' .<_ ) s -> s ( .<_ i^i `' .<_ ) r ) /\ ( ( r ( .<_ i^i `' .<_ ) s /\ s ( .<_ i^i `' .<_ ) t ) -> r ( .<_ i^i `' .<_ ) t ) ) ) |
80 |
79
|
alrimivv |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> A. r A. s A. t ( ( r ( .<_ i^i `' .<_ ) s -> s ( .<_ i^i `' .<_ ) r ) /\ ( ( r ( .<_ i^i `' .<_ ) s /\ s ( .<_ i^i `' .<_ ) t ) -> r ( .<_ i^i `' .<_ ) t ) ) ) |
81 |
|
dfer2 |
|- ( ( .<_ i^i `' .<_ ) Er dom ( .<_ i^i `' .<_ ) <-> ( Rel ( .<_ i^i `' .<_ ) /\ dom ( .<_ i^i `' .<_ ) = dom ( .<_ i^i `' .<_ ) /\ A. r A. s A. t ( ( r ( .<_ i^i `' .<_ ) s -> s ( .<_ i^i `' .<_ ) r ) /\ ( ( r ( .<_ i^i `' .<_ ) s /\ s ( .<_ i^i `' .<_ ) t ) -> r ( .<_ i^i `' .<_ ) t ) ) ) ) |
82 |
5 6 80 81
|
syl3anbrc |
|- ( A. a A. b A. c ( ( a .<_ b /\ b .<_ c ) -> a .<_ c ) -> ( .<_ i^i `' .<_ ) Er dom ( .<_ i^i `' .<_ ) ) |