Step |
Hyp |
Ref |
Expression |
1 |
|
leweon.1 |
|- L = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) } |
2 |
|
r0weon.1 |
|- R = { <. z , w >. | ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) /\ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) e. ( ( 1st ` w ) u. ( 2nd ` w ) ) \/ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) /\ z L w ) ) ) } |
3 |
|
fveq2 |
|- ( x = z -> ( 1st ` x ) = ( 1st ` z ) ) |
4 |
|
fveq2 |
|- ( x = z -> ( 2nd ` x ) = ( 2nd ` z ) ) |
5 |
3 4
|
uneq12d |
|- ( x = z -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) ) |
6 |
|
eqid |
|- ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
7 |
|
fvex |
|- ( 1st ` z ) e. _V |
8 |
|
fvex |
|- ( 2nd ` z ) e. _V |
9 |
7 8
|
unex |
|- ( ( 1st ` z ) u. ( 2nd ` z ) ) e. _V |
10 |
5 6 9
|
fvmpt |
|- ( z e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( 1st ` z ) u. ( 2nd ` z ) ) ) |
11 |
|
fveq2 |
|- ( x = w -> ( 1st ` x ) = ( 1st ` w ) ) |
12 |
|
fveq2 |
|- ( x = w -> ( 2nd ` x ) = ( 2nd ` w ) ) |
13 |
11 12
|
uneq12d |
|- ( x = w -> ( ( 1st ` x ) u. ( 2nd ` x ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) ) |
14 |
|
fvex |
|- ( 1st ` w ) e. _V |
15 |
|
fvex |
|- ( 2nd ` w ) e. _V |
16 |
14 15
|
unex |
|- ( ( 1st ` w ) u. ( 2nd ` w ) ) e. _V |
17 |
13 6 16
|
fvmpt |
|- ( w e. ( On X. On ) -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) ) |
18 |
10 17
|
breqan12d |
|- ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) -> ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) _E ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) <-> ( ( 1st ` z ) u. ( 2nd ` z ) ) _E ( ( 1st ` w ) u. ( 2nd ` w ) ) ) ) |
19 |
16
|
epeli |
|- ( ( ( 1st ` z ) u. ( 2nd ` z ) ) _E ( ( 1st ` w ) u. ( 2nd ` w ) ) <-> ( ( 1st ` z ) u. ( 2nd ` z ) ) e. ( ( 1st ` w ) u. ( 2nd ` w ) ) ) |
20 |
18 19
|
bitrdi |
|- ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) -> ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) _E ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) <-> ( ( 1st ` z ) u. ( 2nd ` z ) ) e. ( ( 1st ` w ) u. ( 2nd ` w ) ) ) ) |
21 |
10 17
|
eqeqan12d |
|- ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) -> ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) <-> ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) ) ) |
22 |
21
|
anbi1d |
|- ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) -> ( ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) /\ z L w ) <-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) /\ z L w ) ) ) |
23 |
20 22
|
orbi12d |
|- ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) -> ( ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) _E ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) \/ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) /\ z L w ) ) <-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) e. ( ( 1st ` w ) u. ( 2nd ` w ) ) \/ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) /\ z L w ) ) ) ) |
24 |
23
|
pm5.32i |
|- ( ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) /\ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) _E ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) \/ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) /\ z L w ) ) ) <-> ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) /\ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) e. ( ( 1st ` w ) u. ( 2nd ` w ) ) \/ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) /\ z L w ) ) ) ) |
25 |
24
|
opabbii |
|- { <. z , w >. | ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) /\ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) _E ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) \/ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) /\ z L w ) ) ) } = { <. z , w >. | ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) /\ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) e. ( ( 1st ` w ) u. ( 2nd ` w ) ) \/ ( ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( ( 1st ` w ) u. ( 2nd ` w ) ) /\ z L w ) ) ) } |
26 |
2 25
|
eqtr4i |
|- R = { <. z , w >. | ( ( z e. ( On X. On ) /\ w e. ( On X. On ) ) /\ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) _E ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) \/ ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` z ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ` w ) /\ z L w ) ) ) } |
27 |
|
xp1st |
|- ( x e. ( On X. On ) -> ( 1st ` x ) e. On ) |
28 |
|
xp2nd |
|- ( x e. ( On X. On ) -> ( 2nd ` x ) e. On ) |
29 |
|
fvex |
|- ( 1st ` x ) e. _V |
30 |
29
|
elon |
|- ( ( 1st ` x ) e. On <-> Ord ( 1st ` x ) ) |
31 |
|
fvex |
|- ( 2nd ` x ) e. _V |
32 |
31
|
elon |
|- ( ( 2nd ` x ) e. On <-> Ord ( 2nd ` x ) ) |
33 |
|
ordun |
|- ( ( Ord ( 1st ` x ) /\ Ord ( 2nd ` x ) ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
34 |
30 32 33
|
syl2anb |
|- ( ( ( 1st ` x ) e. On /\ ( 2nd ` x ) e. On ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
35 |
27 28 34
|
syl2anc |
|- ( x e. ( On X. On ) -> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
36 |
29 31
|
unex |
|- ( ( 1st ` x ) u. ( 2nd ` x ) ) e. _V |
37 |
36
|
elon |
|- ( ( ( 1st ` x ) u. ( 2nd ` x ) ) e. On <-> Ord ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
38 |
35 37
|
sylibr |
|- ( x e. ( On X. On ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. On ) |
39 |
6 38
|
fmpti |
|- ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) : ( On X. On ) --> On |
40 |
39
|
a1i |
|- ( T. -> ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) : ( On X. On ) --> On ) |
41 |
|
epweon |
|- _E We On |
42 |
41
|
a1i |
|- ( T. -> _E We On ) |
43 |
1
|
leweon |
|- L We ( On X. On ) |
44 |
43
|
a1i |
|- ( T. -> L We ( On X. On ) ) |
45 |
|
vex |
|- u e. _V |
46 |
45
|
dmex |
|- dom u e. _V |
47 |
45
|
rnex |
|- ran u e. _V |
48 |
46 47
|
unex |
|- ( dom u u. ran u ) e. _V |
49 |
|
imadmres |
|- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) ) = ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) |
50 |
|
inss2 |
|- ( u i^i ( On X. On ) ) C_ ( On X. On ) |
51 |
|
ssun1 |
|- dom u C_ ( dom u u. ran u ) |
52 |
|
elinel2 |
|- ( x e. ( u i^i ( On X. On ) ) -> x e. ( On X. On ) ) |
53 |
|
1st2nd2 |
|- ( x e. ( On X. On ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
54 |
52 53
|
syl |
|- ( x e. ( u i^i ( On X. On ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
55 |
|
elinel1 |
|- ( x e. ( u i^i ( On X. On ) ) -> x e. u ) |
56 |
54 55
|
eqeltrrd |
|- ( x e. ( u i^i ( On X. On ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. u ) |
57 |
29 31
|
opeldm |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. u -> ( 1st ` x ) e. dom u ) |
58 |
56 57
|
syl |
|- ( x e. ( u i^i ( On X. On ) ) -> ( 1st ` x ) e. dom u ) |
59 |
51 58
|
sselid |
|- ( x e. ( u i^i ( On X. On ) ) -> ( 1st ` x ) e. ( dom u u. ran u ) ) |
60 |
|
ssun2 |
|- ran u C_ ( dom u u. ran u ) |
61 |
29 31
|
opelrn |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. u -> ( 2nd ` x ) e. ran u ) |
62 |
56 61
|
syl |
|- ( x e. ( u i^i ( On X. On ) ) -> ( 2nd ` x ) e. ran u ) |
63 |
60 62
|
sselid |
|- ( x e. ( u i^i ( On X. On ) ) -> ( 2nd ` x ) e. ( dom u u. ran u ) ) |
64 |
59 63
|
prssd |
|- ( x e. ( u i^i ( On X. On ) ) -> { ( 1st ` x ) , ( 2nd ` x ) } C_ ( dom u u. ran u ) ) |
65 |
52 27
|
syl |
|- ( x e. ( u i^i ( On X. On ) ) -> ( 1st ` x ) e. On ) |
66 |
52 28
|
syl |
|- ( x e. ( u i^i ( On X. On ) ) -> ( 2nd ` x ) e. On ) |
67 |
|
ordunpr |
|- ( ( ( 1st ` x ) e. On /\ ( 2nd ` x ) e. On ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. { ( 1st ` x ) , ( 2nd ` x ) } ) |
68 |
65 66 67
|
syl2anc |
|- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. { ( 1st ` x ) , ( 2nd ` x ) } ) |
69 |
64 68
|
sseldd |
|- ( x e. ( u i^i ( On X. On ) ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) ) |
70 |
69
|
rgen |
|- A. x e. ( u i^i ( On X. On ) ) ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) |
71 |
|
ssrab |
|- ( ( u i^i ( On X. On ) ) C_ { x e. ( On X. On ) | ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) } <-> ( ( u i^i ( On X. On ) ) C_ ( On X. On ) /\ A. x e. ( u i^i ( On X. On ) ) ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) ) ) |
72 |
50 70 71
|
mpbir2an |
|- ( u i^i ( On X. On ) ) C_ { x e. ( On X. On ) | ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) } |
73 |
|
dmres |
|- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) = ( u i^i dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) |
74 |
39
|
fdmi |
|- dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) = ( On X. On ) |
75 |
74
|
ineq2i |
|- ( u i^i dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) = ( u i^i ( On X. On ) ) |
76 |
73 75
|
eqtri |
|- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) = ( u i^i ( On X. On ) ) |
77 |
6
|
mptpreima |
|- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " ( dom u u. ran u ) ) = { x e. ( On X. On ) | ( ( 1st ` x ) u. ( 2nd ` x ) ) e. ( dom u u. ran u ) } |
78 |
72 76 77
|
3sstr4i |
|- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " ( dom u u. ran u ) ) |
79 |
|
funmpt |
|- Fun ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
80 |
|
resss |
|- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
81 |
|
dmss |
|- ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) -> dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) |
82 |
80 81
|
ax-mp |
|- dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |
83 |
|
funimass3 |
|- ( ( Fun ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) /\ dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ dom ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) ) -> ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) ) C_ ( dom u u. ran u ) <-> dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " ( dom u u. ran u ) ) ) ) |
84 |
79 82 83
|
mp2an |
|- ( ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) ) C_ ( dom u u. ran u ) <-> dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) C_ ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " ( dom u u. ran u ) ) ) |
85 |
78 84
|
mpbir |
|- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " dom ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) |` u ) ) C_ ( dom u u. ran u ) |
86 |
49 85
|
eqsstrri |
|- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( dom u u. ran u ) |
87 |
48 86
|
ssexi |
|- ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) e. _V |
88 |
87
|
a1i |
|- ( T. -> ( ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) e. _V ) |
89 |
26 40 42 44 88
|
fnwe |
|- ( T. -> R We ( On X. On ) ) |
90 |
|
epse |
|- _E Se On |
91 |
90
|
a1i |
|- ( T. -> _E Se On ) |
92 |
|
vuniex |
|- U. u e. _V |
93 |
92
|
pwex |
|- ~P U. u e. _V |
94 |
93 93
|
xpex |
|- ( ~P U. u X. ~P U. u ) e. _V |
95 |
6
|
mptpreima |
|- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) = { x e. ( On X. On ) | ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u } |
96 |
|
df-rab |
|- { x e. ( On X. On ) | ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u } = { x | ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) } |
97 |
95 96
|
eqtri |
|- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) = { x | ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) } |
98 |
53
|
adantr |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
99 |
|
elssuni |
|- ( ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ U. u ) |
100 |
99
|
adantl |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ U. u ) |
101 |
100
|
unssad |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 1st ` x ) C_ U. u ) |
102 |
29
|
elpw |
|- ( ( 1st ` x ) e. ~P U. u <-> ( 1st ` x ) C_ U. u ) |
103 |
101 102
|
sylibr |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 1st ` x ) e. ~P U. u ) |
104 |
100
|
unssbd |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 2nd ` x ) C_ U. u ) |
105 |
31
|
elpw |
|- ( ( 2nd ` x ) e. ~P U. u <-> ( 2nd ` x ) C_ U. u ) |
106 |
104 105
|
sylibr |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( 2nd ` x ) e. ~P U. u ) |
107 |
103 106
|
jca |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> ( ( 1st ` x ) e. ~P U. u /\ ( 2nd ` x ) e. ~P U. u ) ) |
108 |
|
elxp6 |
|- ( x e. ( ~P U. u X. ~P U. u ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ~P U. u /\ ( 2nd ` x ) e. ~P U. u ) ) ) |
109 |
98 107 108
|
sylanbrc |
|- ( ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) -> x e. ( ~P U. u X. ~P U. u ) ) |
110 |
109
|
abssi |
|- { x | ( x e. ( On X. On ) /\ ( ( 1st ` x ) u. ( 2nd ` x ) ) e. u ) } C_ ( ~P U. u X. ~P U. u ) |
111 |
97 110
|
eqsstri |
|- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) C_ ( ~P U. u X. ~P U. u ) |
112 |
94 111
|
ssexi |
|- ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) e. _V |
113 |
112
|
a1i |
|- ( T. -> ( `' ( x e. ( On X. On ) |-> ( ( 1st ` x ) u. ( 2nd ` x ) ) ) " u ) e. _V ) |
114 |
26 40 91 113
|
fnse |
|- ( T. -> R Se ( On X. On ) ) |
115 |
89 114
|
jca |
|- ( T. -> ( R We ( On X. On ) /\ R Se ( On X. On ) ) ) |
116 |
115
|
mptru |
|- ( R We ( On X. On ) /\ R Se ( On X. On ) ) |