Step |
Hyp |
Ref |
Expression |
1 |
|
isfi |
|- ( z e. Fin <-> E. w e. _om z ~~ w ) |
2 |
|
nnfi |
|- ( w e. _om -> w e. Fin ) |
3 |
|
ensymfib |
|- ( w e. Fin -> ( w ~~ z <-> z ~~ w ) ) |
4 |
2 3
|
syl |
|- ( w e. _om -> ( w ~~ z <-> z ~~ w ) ) |
5 |
|
breq1 |
|- ( w = (/) -> ( w ~~ z <-> (/) ~~ z ) ) |
6 |
5
|
anbi2d |
|- ( w = (/) -> ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) <-> ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) ) ) |
7 |
6
|
imbi1d |
|- ( w = (/) -> ( ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) <-> ( ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) -> |^| z e. A ) ) ) |
8 |
7
|
albidv |
|- ( w = (/) -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) <-> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) -> |^| z e. A ) ) ) |
9 |
|
breq1 |
|- ( w = t -> ( w ~~ z <-> t ~~ z ) ) |
10 |
9
|
anbi2d |
|- ( w = t -> ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) <-> ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) ) ) |
11 |
10
|
imbi1d |
|- ( w = t -> ( ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) <-> ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) ) ) |
12 |
11
|
albidv |
|- ( w = t -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) <-> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) ) ) |
13 |
|
breq1 |
|- ( w = suc t -> ( w ~~ z <-> suc t ~~ z ) ) |
14 |
13
|
anbi2d |
|- ( w = suc t -> ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) <-> ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) ) ) |
15 |
14
|
imbi1d |
|- ( w = suc t -> ( ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) <-> ( ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) -> |^| z e. A ) ) ) |
16 |
15
|
albidv |
|- ( w = suc t -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) <-> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) -> |^| z e. A ) ) ) |
17 |
|
en0r |
|- ( (/) ~~ z <-> z = (/) ) |
18 |
17
|
biimpi |
|- ( (/) ~~ z -> z = (/) ) |
19 |
18
|
anim1i |
|- ( ( (/) ~~ z /\ z =/= (/) ) -> ( z = (/) /\ z =/= (/) ) ) |
20 |
19
|
ancoms |
|- ( ( z =/= (/) /\ (/) ~~ z ) -> ( z = (/) /\ z =/= (/) ) ) |
21 |
20
|
adantll |
|- ( ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) -> ( z = (/) /\ z =/= (/) ) ) |
22 |
|
df-ne |
|- ( z =/= (/) <-> -. z = (/) ) |
23 |
|
pm3.24 |
|- -. ( z = (/) /\ -. z = (/) ) |
24 |
23
|
pm2.21i |
|- ( ( z = (/) /\ -. z = (/) ) -> |^| z e. A ) |
25 |
22 24
|
sylan2b |
|- ( ( z = (/) /\ z =/= (/) ) -> |^| z e. A ) |
26 |
21 25
|
syl |
|- ( ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) -> |^| z e. A ) |
27 |
26
|
ax-gen |
|- A. z ( ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) -> |^| z e. A ) |
28 |
27
|
a1i |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ (/) ~~ z ) -> |^| z e. A ) ) |
29 |
|
nfv |
|- F/ z A. v e. A A. u e. A ( v i^i u ) e. A |
30 |
|
nfa1 |
|- F/ z A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) |
31 |
|
bren |
|- ( suc t ~~ z <-> E. f f : suc t -1-1-onto-> z ) |
32 |
|
ssel |
|- ( z C_ A -> ( ( f ` t ) e. z -> ( f ` t ) e. A ) ) |
33 |
|
f1of |
|- ( f : suc t -1-1-onto-> z -> f : suc t --> z ) |
34 |
|
vex |
|- t e. _V |
35 |
34
|
sucid |
|- t e. suc t |
36 |
|
ffvelcdm |
|- ( ( f : suc t --> z /\ t e. suc t ) -> ( f ` t ) e. z ) |
37 |
33 35 36
|
sylancl |
|- ( f : suc t -1-1-onto-> z -> ( f ` t ) e. z ) |
38 |
32 37
|
impel |
|- ( ( z C_ A /\ f : suc t -1-1-onto-> z ) -> ( f ` t ) e. A ) |
39 |
38
|
adantr |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> ( f ` t ) e. A ) |
40 |
|
df-ne |
|- ( ( f " t ) =/= (/) <-> -. ( f " t ) = (/) ) |
41 |
|
imassrn |
|- ( f " t ) C_ ran f |
42 |
|
dff1o2 |
|- ( f : suc t -1-1-onto-> z <-> ( f Fn suc t /\ Fun `' f /\ ran f = z ) ) |
43 |
42
|
simp3bi |
|- ( f : suc t -1-1-onto-> z -> ran f = z ) |
44 |
41 43
|
sseqtrid |
|- ( f : suc t -1-1-onto-> z -> ( f " t ) C_ z ) |
45 |
|
sstr2 |
|- ( ( f " t ) C_ z -> ( z C_ A -> ( f " t ) C_ A ) ) |
46 |
44 45
|
syl |
|- ( f : suc t -1-1-onto-> z -> ( z C_ A -> ( f " t ) C_ A ) ) |
47 |
46
|
anim1d |
|- ( f : suc t -1-1-onto-> z -> ( ( z C_ A /\ ( f " t ) =/= (/) ) -> ( ( f " t ) C_ A /\ ( f " t ) =/= (/) ) ) ) |
48 |
|
f1of1 |
|- ( f : suc t -1-1-onto-> z -> f : suc t -1-1-> z ) |
49 |
|
sssucid |
|- t C_ suc t |
50 |
|
vex |
|- f e. _V |
51 |
|
f1imaen3g |
|- ( ( f : suc t -1-1-> z /\ t C_ suc t /\ f e. _V ) -> t ~~ ( f " t ) ) |
52 |
49 50 51
|
mp3an23 |
|- ( f : suc t -1-1-> z -> t ~~ ( f " t ) ) |
53 |
48 52
|
syl |
|- ( f : suc t -1-1-onto-> z -> t ~~ ( f " t ) ) |
54 |
47 53
|
jctird |
|- ( f : suc t -1-1-onto-> z -> ( ( z C_ A /\ ( f " t ) =/= (/) ) -> ( ( ( f " t ) C_ A /\ ( f " t ) =/= (/) ) /\ t ~~ ( f " t ) ) ) ) |
55 |
50
|
imaex |
|- ( f " t ) e. _V |
56 |
|
sseq1 |
|- ( z = ( f " t ) -> ( z C_ A <-> ( f " t ) C_ A ) ) |
57 |
|
neeq1 |
|- ( z = ( f " t ) -> ( z =/= (/) <-> ( f " t ) =/= (/) ) ) |
58 |
56 57
|
anbi12d |
|- ( z = ( f " t ) -> ( ( z C_ A /\ z =/= (/) ) <-> ( ( f " t ) C_ A /\ ( f " t ) =/= (/) ) ) ) |
59 |
|
breq2 |
|- ( z = ( f " t ) -> ( t ~~ z <-> t ~~ ( f " t ) ) ) |
60 |
58 59
|
anbi12d |
|- ( z = ( f " t ) -> ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) <-> ( ( ( f " t ) C_ A /\ ( f " t ) =/= (/) ) /\ t ~~ ( f " t ) ) ) ) |
61 |
|
inteq |
|- ( z = ( f " t ) -> |^| z = |^| ( f " t ) ) |
62 |
61
|
eleq1d |
|- ( z = ( f " t ) -> ( |^| z e. A <-> |^| ( f " t ) e. A ) ) |
63 |
60 62
|
imbi12d |
|- ( z = ( f " t ) -> ( ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) <-> ( ( ( ( f " t ) C_ A /\ ( f " t ) =/= (/) ) /\ t ~~ ( f " t ) ) -> |^| ( f " t ) e. A ) ) ) |
64 |
55 63
|
spcv |
|- ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( ( ( ( f " t ) C_ A /\ ( f " t ) =/= (/) ) /\ t ~~ ( f " t ) ) -> |^| ( f " t ) e. A ) ) |
65 |
54 64
|
sylan9 |
|- ( ( f : suc t -1-1-onto-> z /\ A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) ) -> ( ( z C_ A /\ ( f " t ) =/= (/) ) -> |^| ( f " t ) e. A ) ) |
66 |
|
ineq1 |
|- ( v = |^| ( f " t ) -> ( v i^i u ) = ( |^| ( f " t ) i^i u ) ) |
67 |
66
|
eleq1d |
|- ( v = |^| ( f " t ) -> ( ( v i^i u ) e. A <-> ( |^| ( f " t ) i^i u ) e. A ) ) |
68 |
|
ineq2 |
|- ( u = ( f ` t ) -> ( |^| ( f " t ) i^i u ) = ( |^| ( f " t ) i^i ( f ` t ) ) ) |
69 |
68
|
eleq1d |
|- ( u = ( f ` t ) -> ( ( |^| ( f " t ) i^i u ) e. A <-> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) |
70 |
67 69
|
rspc2v |
|- ( ( |^| ( f " t ) e. A /\ ( f ` t ) e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) |
71 |
70
|
ex |
|- ( |^| ( f " t ) e. A -> ( ( f ` t ) e. A -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) |
72 |
65 71
|
syl6 |
|- ( ( f : suc t -1-1-onto-> z /\ A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) ) -> ( ( z C_ A /\ ( f " t ) =/= (/) ) -> ( ( f ` t ) e. A -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) ) |
73 |
72
|
com4r |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( ( f : suc t -1-1-onto-> z /\ A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) ) -> ( ( z C_ A /\ ( f " t ) =/= (/) ) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) ) |
74 |
73
|
exp5c |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( f : suc t -1-1-onto-> z -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( z C_ A -> ( ( f " t ) =/= (/) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) ) ) ) |
75 |
74
|
com14 |
|- ( z C_ A -> ( f : suc t -1-1-onto-> z -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( ( f " t ) =/= (/) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) ) ) ) |
76 |
75
|
imp43 |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> ( ( f " t ) =/= (/) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) |
77 |
40 76
|
biimtrrid |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> ( -. ( f " t ) = (/) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) ) |
78 |
|
inteq |
|- ( ( f " t ) = (/) -> |^| ( f " t ) = |^| (/) ) |
79 |
|
int0 |
|- |^| (/) = _V |
80 |
78 79
|
eqtrdi |
|- ( ( f " t ) = (/) -> |^| ( f " t ) = _V ) |
81 |
80
|
ineq1d |
|- ( ( f " t ) = (/) -> ( |^| ( f " t ) i^i ( f ` t ) ) = ( _V i^i ( f ` t ) ) ) |
82 |
|
ssv |
|- ( f ` t ) C_ _V |
83 |
|
sseqin2 |
|- ( ( f ` t ) C_ _V <-> ( _V i^i ( f ` t ) ) = ( f ` t ) ) |
84 |
82 83
|
mpbi |
|- ( _V i^i ( f ` t ) ) = ( f ` t ) |
85 |
81 84
|
eqtrdi |
|- ( ( f " t ) = (/) -> ( |^| ( f " t ) i^i ( f ` t ) ) = ( f ` t ) ) |
86 |
85
|
eleq1d |
|- ( ( f " t ) = (/) -> ( ( |^| ( f " t ) i^i ( f ` t ) ) e. A <-> ( f ` t ) e. A ) ) |
87 |
86
|
biimprd |
|- ( ( f " t ) = (/) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) |
88 |
77 87
|
pm2.61d2 |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> ( ( f ` t ) e. A -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) ) |
89 |
39 88
|
mpd |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> ( |^| ( f " t ) i^i ( f ` t ) ) e. A ) |
90 |
|
fvex |
|- ( f ` t ) e. _V |
91 |
90
|
intunsn |
|- |^| ( ( f " t ) u. { ( f ` t ) } ) = ( |^| ( f " t ) i^i ( f ` t ) ) |
92 |
|
f1ofn |
|- ( f : suc t -1-1-onto-> z -> f Fn suc t ) |
93 |
|
fnsnfv |
|- ( ( f Fn suc t /\ t e. suc t ) -> { ( f ` t ) } = ( f " { t } ) ) |
94 |
92 35 93
|
sylancl |
|- ( f : suc t -1-1-onto-> z -> { ( f ` t ) } = ( f " { t } ) ) |
95 |
94
|
uneq2d |
|- ( f : suc t -1-1-onto-> z -> ( ( f " t ) u. { ( f ` t ) } ) = ( ( f " t ) u. ( f " { t } ) ) ) |
96 |
|
df-suc |
|- suc t = ( t u. { t } ) |
97 |
96
|
imaeq2i |
|- ( f " suc t ) = ( f " ( t u. { t } ) ) |
98 |
|
imaundi |
|- ( f " ( t u. { t } ) ) = ( ( f " t ) u. ( f " { t } ) ) |
99 |
97 98
|
eqtr2i |
|- ( ( f " t ) u. ( f " { t } ) ) = ( f " suc t ) |
100 |
95 99
|
eqtrdi |
|- ( f : suc t -1-1-onto-> z -> ( ( f " t ) u. { ( f ` t ) } ) = ( f " suc t ) ) |
101 |
|
f1ofo |
|- ( f : suc t -1-1-onto-> z -> f : suc t -onto-> z ) |
102 |
|
foima |
|- ( f : suc t -onto-> z -> ( f " suc t ) = z ) |
103 |
101 102
|
syl |
|- ( f : suc t -1-1-onto-> z -> ( f " suc t ) = z ) |
104 |
100 103
|
eqtrd |
|- ( f : suc t -1-1-onto-> z -> ( ( f " t ) u. { ( f ` t ) } ) = z ) |
105 |
104
|
inteqd |
|- ( f : suc t -1-1-onto-> z -> |^| ( ( f " t ) u. { ( f ` t ) } ) = |^| z ) |
106 |
91 105
|
eqtr3id |
|- ( f : suc t -1-1-onto-> z -> ( |^| ( f " t ) i^i ( f ` t ) ) = |^| z ) |
107 |
106
|
eleq1d |
|- ( f : suc t -1-1-onto-> z -> ( ( |^| ( f " t ) i^i ( f ` t ) ) e. A <-> |^| z e. A ) ) |
108 |
107
|
ad2antlr |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> ( ( |^| ( f " t ) i^i ( f ` t ) ) e. A <-> |^| z e. A ) ) |
109 |
89 108
|
mpbid |
|- ( ( ( z C_ A /\ f : suc t -1-1-onto-> z ) /\ ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) /\ A. v e. A A. u e. A ( v i^i u ) e. A ) ) -> |^| z e. A ) |
110 |
109
|
exp43 |
|- ( z C_ A -> ( f : suc t -1-1-onto-> z -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) ) |
111 |
110
|
exlimdv |
|- ( z C_ A -> ( E. f f : suc t -1-1-onto-> z -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) ) |
112 |
31 111
|
biimtrid |
|- ( z C_ A -> ( suc t ~~ z -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) ) |
113 |
112
|
imp |
|- ( ( z C_ A /\ suc t ~~ z ) -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) |
114 |
113
|
adantlr |
|- ( ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) |
115 |
114
|
com13 |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> ( ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) -> |^| z e. A ) ) ) |
116 |
29 30 115
|
alrimd |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) -> |^| z e. A ) ) ) |
117 |
116
|
a1i |
|- ( t e. _om -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ t ~~ z ) -> |^| z e. A ) -> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ suc t ~~ z ) -> |^| z e. A ) ) ) ) |
118 |
8 12 16 28 117
|
finds2 |
|- ( w e. _om -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) ) ) |
119 |
|
sp |
|- ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) -> ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) ) |
120 |
118 119
|
syl6 |
|- ( w e. _om -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( ( ( z C_ A /\ z =/= (/) ) /\ w ~~ z ) -> |^| z e. A ) ) ) |
121 |
120
|
exp4a |
|- ( w e. _om -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( ( z C_ A /\ z =/= (/) ) -> ( w ~~ z -> |^| z e. A ) ) ) ) |
122 |
121
|
com24 |
|- ( w e. _om -> ( w ~~ z -> ( ( z C_ A /\ z =/= (/) ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) ) |
123 |
4 122
|
sylbird |
|- ( w e. _om -> ( z ~~ w -> ( ( z C_ A /\ z =/= (/) ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) ) |
124 |
123
|
rexlimiv |
|- ( E. w e. _om z ~~ w -> ( ( z C_ A /\ z =/= (/) ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) |
125 |
1 124
|
sylbi |
|- ( z e. Fin -> ( ( z C_ A /\ z =/= (/) ) -> ( A. v e. A A. u e. A ( v i^i u ) e. A -> |^| z e. A ) ) ) |
126 |
125
|
com13 |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( ( z C_ A /\ z =/= (/) ) -> ( z e. Fin -> |^| z e. A ) ) ) |
127 |
126
|
impd |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) ) |
128 |
127
|
alrimiv |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A -> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) ) |
129 |
|
zfpair2 |
|- { v , u } e. _V |
130 |
|
sseq1 |
|- ( z = { v , u } -> ( z C_ A <-> { v , u } C_ A ) ) |
131 |
|
neeq1 |
|- ( z = { v , u } -> ( z =/= (/) <-> { v , u } =/= (/) ) ) |
132 |
130 131
|
anbi12d |
|- ( z = { v , u } -> ( ( z C_ A /\ z =/= (/) ) <-> ( { v , u } C_ A /\ { v , u } =/= (/) ) ) ) |
133 |
|
eleq1 |
|- ( z = { v , u } -> ( z e. Fin <-> { v , u } e. Fin ) ) |
134 |
132 133
|
anbi12d |
|- ( z = { v , u } -> ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) <-> ( ( { v , u } C_ A /\ { v , u } =/= (/) ) /\ { v , u } e. Fin ) ) ) |
135 |
|
inteq |
|- ( z = { v , u } -> |^| z = |^| { v , u } ) |
136 |
135
|
eleq1d |
|- ( z = { v , u } -> ( |^| z e. A <-> |^| { v , u } e. A ) ) |
137 |
134 136
|
imbi12d |
|- ( z = { v , u } -> ( ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) <-> ( ( ( { v , u } C_ A /\ { v , u } =/= (/) ) /\ { v , u } e. Fin ) -> |^| { v , u } e. A ) ) ) |
138 |
129 137
|
spcv |
|- ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) -> ( ( ( { v , u } C_ A /\ { v , u } =/= (/) ) /\ { v , u } e. Fin ) -> |^| { v , u } e. A ) ) |
139 |
|
vex |
|- v e. _V |
140 |
|
vex |
|- u e. _V |
141 |
139 140
|
prss |
|- ( ( v e. A /\ u e. A ) <-> { v , u } C_ A ) |
142 |
139
|
prnz |
|- { v , u } =/= (/) |
143 |
142
|
biantru |
|- ( { v , u } C_ A <-> ( { v , u } C_ A /\ { v , u } =/= (/) ) ) |
144 |
|
prfi |
|- { v , u } e. Fin |
145 |
144
|
biantru |
|- ( ( { v , u } C_ A /\ { v , u } =/= (/) ) <-> ( ( { v , u } C_ A /\ { v , u } =/= (/) ) /\ { v , u } e. Fin ) ) |
146 |
141 143 145
|
3bitrri |
|- ( ( ( { v , u } C_ A /\ { v , u } =/= (/) ) /\ { v , u } e. Fin ) <-> ( v e. A /\ u e. A ) ) |
147 |
139 140
|
intpr |
|- |^| { v , u } = ( v i^i u ) |
148 |
147
|
eleq1i |
|- ( |^| { v , u } e. A <-> ( v i^i u ) e. A ) |
149 |
138 146 148
|
3imtr3g |
|- ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) -> ( ( v e. A /\ u e. A ) -> ( v i^i u ) e. A ) ) |
150 |
149
|
ralrimivv |
|- ( A. z ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) -> A. v e. A A. u e. A ( v i^i u ) e. A ) |
151 |
128 150
|
impbii |
|- ( A. v e. A A. u e. A ( v i^i u ) e. A <-> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) ) |
152 |
|
ineq1 |
|- ( x = v -> ( x i^i y ) = ( v i^i y ) ) |
153 |
152
|
eleq1d |
|- ( x = v -> ( ( x i^i y ) e. A <-> ( v i^i y ) e. A ) ) |
154 |
|
ineq2 |
|- ( y = u -> ( v i^i y ) = ( v i^i u ) ) |
155 |
154
|
eleq1d |
|- ( y = u -> ( ( v i^i y ) e. A <-> ( v i^i u ) e. A ) ) |
156 |
153 155
|
cbvral2vw |
|- ( A. x e. A A. y e. A ( x i^i y ) e. A <-> A. v e. A A. u e. A ( v i^i u ) e. A ) |
157 |
|
df-3an |
|- ( ( z C_ A /\ z =/= (/) /\ z e. Fin ) <-> ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) ) |
158 |
157
|
imbi1i |
|- ( ( ( z C_ A /\ z =/= (/) /\ z e. Fin ) -> |^| z e. A ) <-> ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) ) |
159 |
158
|
albii |
|- ( A. z ( ( z C_ A /\ z =/= (/) /\ z e. Fin ) -> |^| z e. A ) <-> A. z ( ( ( z C_ A /\ z =/= (/) ) /\ z e. Fin ) -> |^| z e. A ) ) |
160 |
151 156 159
|
3bitr4i |
|- ( A. x e. A A. y e. A ( x i^i y ) e. A <-> A. z ( ( z C_ A /\ z =/= (/) /\ z e. Fin ) -> |^| z e. A ) ) |