Step |
Hyp |
Ref |
Expression |
1 |
|
isfi |
|- ( x e. Fin <-> E. y e. _om x ~~ y ) |
2 |
|
ensym |
|- ( x ~~ y -> y ~~ x ) |
3 |
|
breq1 |
|- ( y = (/) -> ( y ~~ x <-> (/) ~~ x ) ) |
4 |
3
|
anbi2d |
|- ( y = (/) -> ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) <-> ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) ) ) |
5 |
4
|
imbi1d |
|- ( y = (/) -> ( ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) <-> ( ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) -> |^| x e. A ) ) ) |
6 |
5
|
albidv |
|- ( y = (/) -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) <-> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) -> |^| x e. A ) ) ) |
7 |
|
breq1 |
|- ( y = v -> ( y ~~ x <-> v ~~ x ) ) |
8 |
7
|
anbi2d |
|- ( y = v -> ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) <-> ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) ) ) |
9 |
8
|
imbi1d |
|- ( y = v -> ( ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) <-> ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) ) ) |
10 |
9
|
albidv |
|- ( y = v -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) <-> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) ) ) |
11 |
|
breq1 |
|- ( y = suc v -> ( y ~~ x <-> suc v ~~ x ) ) |
12 |
11
|
anbi2d |
|- ( y = suc v -> ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) <-> ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) ) ) |
13 |
12
|
imbi1d |
|- ( y = suc v -> ( ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) <-> ( ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) -> |^| x e. A ) ) ) |
14 |
13
|
albidv |
|- ( y = suc v -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) <-> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) -> |^| x e. A ) ) ) |
15 |
|
ensym |
|- ( (/) ~~ x -> x ~~ (/) ) |
16 |
|
en0 |
|- ( x ~~ (/) <-> x = (/) ) |
17 |
15 16
|
sylib |
|- ( (/) ~~ x -> x = (/) ) |
18 |
17
|
anim1i |
|- ( ( (/) ~~ x /\ x =/= (/) ) -> ( x = (/) /\ x =/= (/) ) ) |
19 |
18
|
ancoms |
|- ( ( x =/= (/) /\ (/) ~~ x ) -> ( x = (/) /\ x =/= (/) ) ) |
20 |
19
|
adantll |
|- ( ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) -> ( x = (/) /\ x =/= (/) ) ) |
21 |
|
df-ne |
|- ( x =/= (/) <-> -. x = (/) ) |
22 |
|
pm3.24 |
|- -. ( x = (/) /\ -. x = (/) ) |
23 |
22
|
pm2.21i |
|- ( ( x = (/) /\ -. x = (/) ) -> |^| x e. A ) |
24 |
21 23
|
sylan2b |
|- ( ( x = (/) /\ x =/= (/) ) -> |^| x e. A ) |
25 |
20 24
|
syl |
|- ( ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) -> |^| x e. A ) |
26 |
25
|
ax-gen |
|- A. x ( ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) -> |^| x e. A ) |
27 |
26
|
a1i |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ (/) ~~ x ) -> |^| x e. A ) ) |
28 |
|
nfv |
|- F/ x A. z e. A A. w e. A ( z i^i w ) e. A |
29 |
|
nfa1 |
|- F/ x A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) |
30 |
|
bren |
|- ( suc v ~~ x <-> E. f f : suc v -1-1-onto-> x ) |
31 |
|
ssel |
|- ( x C_ A -> ( ( f ` v ) e. x -> ( f ` v ) e. A ) ) |
32 |
|
f1of |
|- ( f : suc v -1-1-onto-> x -> f : suc v --> x ) |
33 |
|
vex |
|- v e. _V |
34 |
33
|
sucid |
|- v e. suc v |
35 |
|
ffvelrn |
|- ( ( f : suc v --> x /\ v e. suc v ) -> ( f ` v ) e. x ) |
36 |
32 34 35
|
sylancl |
|- ( f : suc v -1-1-onto-> x -> ( f ` v ) e. x ) |
37 |
31 36
|
impel |
|- ( ( x C_ A /\ f : suc v -1-1-onto-> x ) -> ( f ` v ) e. A ) |
38 |
37
|
adantr |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> ( f ` v ) e. A ) |
39 |
|
df-ne |
|- ( ( f " v ) =/= (/) <-> -. ( f " v ) = (/) ) |
40 |
|
imassrn |
|- ( f " v ) C_ ran f |
41 |
|
dff1o2 |
|- ( f : suc v -1-1-onto-> x <-> ( f Fn suc v /\ Fun `' f /\ ran f = x ) ) |
42 |
41
|
simp3bi |
|- ( f : suc v -1-1-onto-> x -> ran f = x ) |
43 |
40 42
|
sseqtrid |
|- ( f : suc v -1-1-onto-> x -> ( f " v ) C_ x ) |
44 |
|
sstr2 |
|- ( ( f " v ) C_ x -> ( x C_ A -> ( f " v ) C_ A ) ) |
45 |
43 44
|
syl |
|- ( f : suc v -1-1-onto-> x -> ( x C_ A -> ( f " v ) C_ A ) ) |
46 |
45
|
anim1d |
|- ( f : suc v -1-1-onto-> x -> ( ( x C_ A /\ ( f " v ) =/= (/) ) -> ( ( f " v ) C_ A /\ ( f " v ) =/= (/) ) ) ) |
47 |
|
f1of1 |
|- ( f : suc v -1-1-onto-> x -> f : suc v -1-1-> x ) |
48 |
|
vex |
|- x e. _V |
49 |
|
sssucid |
|- v C_ suc v |
50 |
|
f1imaen2g |
|- ( ( ( f : suc v -1-1-> x /\ x e. _V ) /\ ( v C_ suc v /\ v e. _V ) ) -> ( f " v ) ~~ v ) |
51 |
49 33 50
|
mpanr12 |
|- ( ( f : suc v -1-1-> x /\ x e. _V ) -> ( f " v ) ~~ v ) |
52 |
47 48 51
|
sylancl |
|- ( f : suc v -1-1-onto-> x -> ( f " v ) ~~ v ) |
53 |
52
|
ensymd |
|- ( f : suc v -1-1-onto-> x -> v ~~ ( f " v ) ) |
54 |
46 53
|
jctird |
|- ( f : suc v -1-1-onto-> x -> ( ( x C_ A /\ ( f " v ) =/= (/) ) -> ( ( ( f " v ) C_ A /\ ( f " v ) =/= (/) ) /\ v ~~ ( f " v ) ) ) ) |
55 |
|
vex |
|- f e. _V |
56 |
55
|
imaex |
|- ( f " v ) e. _V |
57 |
|
sseq1 |
|- ( x = ( f " v ) -> ( x C_ A <-> ( f " v ) C_ A ) ) |
58 |
|
neeq1 |
|- ( x = ( f " v ) -> ( x =/= (/) <-> ( f " v ) =/= (/) ) ) |
59 |
57 58
|
anbi12d |
|- ( x = ( f " v ) -> ( ( x C_ A /\ x =/= (/) ) <-> ( ( f " v ) C_ A /\ ( f " v ) =/= (/) ) ) ) |
60 |
|
breq2 |
|- ( x = ( f " v ) -> ( v ~~ x <-> v ~~ ( f " v ) ) ) |
61 |
59 60
|
anbi12d |
|- ( x = ( f " v ) -> ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) <-> ( ( ( f " v ) C_ A /\ ( f " v ) =/= (/) ) /\ v ~~ ( f " v ) ) ) ) |
62 |
|
inteq |
|- ( x = ( f " v ) -> |^| x = |^| ( f " v ) ) |
63 |
62
|
eleq1d |
|- ( x = ( f " v ) -> ( |^| x e. A <-> |^| ( f " v ) e. A ) ) |
64 |
61 63
|
imbi12d |
|- ( x = ( f " v ) -> ( ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) <-> ( ( ( ( f " v ) C_ A /\ ( f " v ) =/= (/) ) /\ v ~~ ( f " v ) ) -> |^| ( f " v ) e. A ) ) ) |
65 |
56 64
|
spcv |
|- ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( ( ( ( f " v ) C_ A /\ ( f " v ) =/= (/) ) /\ v ~~ ( f " v ) ) -> |^| ( f " v ) e. A ) ) |
66 |
54 65
|
sylan9 |
|- ( ( f : suc v -1-1-onto-> x /\ A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) ) -> ( ( x C_ A /\ ( f " v ) =/= (/) ) -> |^| ( f " v ) e. A ) ) |
67 |
|
ineq1 |
|- ( z = |^| ( f " v ) -> ( z i^i w ) = ( |^| ( f " v ) i^i w ) ) |
68 |
67
|
eleq1d |
|- ( z = |^| ( f " v ) -> ( ( z i^i w ) e. A <-> ( |^| ( f " v ) i^i w ) e. A ) ) |
69 |
|
ineq2 |
|- ( w = ( f ` v ) -> ( |^| ( f " v ) i^i w ) = ( |^| ( f " v ) i^i ( f ` v ) ) ) |
70 |
69
|
eleq1d |
|- ( w = ( f ` v ) -> ( ( |^| ( f " v ) i^i w ) e. A <-> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) |
71 |
68 70
|
rspc2v |
|- ( ( |^| ( f " v ) e. A /\ ( f ` v ) e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) |
72 |
71
|
ex |
|- ( |^| ( f " v ) e. A -> ( ( f ` v ) e. A -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) |
73 |
66 72
|
syl6 |
|- ( ( f : suc v -1-1-onto-> x /\ A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) ) -> ( ( x C_ A /\ ( f " v ) =/= (/) ) -> ( ( f ` v ) e. A -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) ) |
74 |
73
|
com4r |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( ( f : suc v -1-1-onto-> x /\ A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) ) -> ( ( x C_ A /\ ( f " v ) =/= (/) ) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) ) |
75 |
74
|
exp5c |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( f : suc v -1-1-onto-> x -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( x C_ A -> ( ( f " v ) =/= (/) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) ) ) ) |
76 |
75
|
com14 |
|- ( x C_ A -> ( f : suc v -1-1-onto-> x -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( ( f " v ) =/= (/) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) ) ) ) |
77 |
76
|
imp43 |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> ( ( f " v ) =/= (/) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) |
78 |
39 77
|
syl5bir |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> ( -. ( f " v ) = (/) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) ) |
79 |
|
inteq |
|- ( ( f " v ) = (/) -> |^| ( f " v ) = |^| (/) ) |
80 |
|
int0 |
|- |^| (/) = _V |
81 |
79 80
|
eqtrdi |
|- ( ( f " v ) = (/) -> |^| ( f " v ) = _V ) |
82 |
81
|
ineq1d |
|- ( ( f " v ) = (/) -> ( |^| ( f " v ) i^i ( f ` v ) ) = ( _V i^i ( f ` v ) ) ) |
83 |
|
ssv |
|- ( f ` v ) C_ _V |
84 |
|
sseqin2 |
|- ( ( f ` v ) C_ _V <-> ( _V i^i ( f ` v ) ) = ( f ` v ) ) |
85 |
83 84
|
mpbi |
|- ( _V i^i ( f ` v ) ) = ( f ` v ) |
86 |
82 85
|
eqtrdi |
|- ( ( f " v ) = (/) -> ( |^| ( f " v ) i^i ( f ` v ) ) = ( f ` v ) ) |
87 |
86
|
eleq1d |
|- ( ( f " v ) = (/) -> ( ( |^| ( f " v ) i^i ( f ` v ) ) e. A <-> ( f ` v ) e. A ) ) |
88 |
87
|
biimprd |
|- ( ( f " v ) = (/) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) |
89 |
78 88
|
pm2.61d2 |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> ( ( f ` v ) e. A -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) ) |
90 |
38 89
|
mpd |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> ( |^| ( f " v ) i^i ( f ` v ) ) e. A ) |
91 |
|
fvex |
|- ( f ` v ) e. _V |
92 |
91
|
intunsn |
|- |^| ( ( f " v ) u. { ( f ` v ) } ) = ( |^| ( f " v ) i^i ( f ` v ) ) |
93 |
|
f1ofn |
|- ( f : suc v -1-1-onto-> x -> f Fn suc v ) |
94 |
|
fnsnfv |
|- ( ( f Fn suc v /\ v e. suc v ) -> { ( f ` v ) } = ( f " { v } ) ) |
95 |
93 34 94
|
sylancl |
|- ( f : suc v -1-1-onto-> x -> { ( f ` v ) } = ( f " { v } ) ) |
96 |
95
|
uneq2d |
|- ( f : suc v -1-1-onto-> x -> ( ( f " v ) u. { ( f ` v ) } ) = ( ( f " v ) u. ( f " { v } ) ) ) |
97 |
|
df-suc |
|- suc v = ( v u. { v } ) |
98 |
97
|
imaeq2i |
|- ( f " suc v ) = ( f " ( v u. { v } ) ) |
99 |
|
imaundi |
|- ( f " ( v u. { v } ) ) = ( ( f " v ) u. ( f " { v } ) ) |
100 |
98 99
|
eqtr2i |
|- ( ( f " v ) u. ( f " { v } ) ) = ( f " suc v ) |
101 |
96 100
|
eqtrdi |
|- ( f : suc v -1-1-onto-> x -> ( ( f " v ) u. { ( f ` v ) } ) = ( f " suc v ) ) |
102 |
|
f1ofo |
|- ( f : suc v -1-1-onto-> x -> f : suc v -onto-> x ) |
103 |
|
foima |
|- ( f : suc v -onto-> x -> ( f " suc v ) = x ) |
104 |
102 103
|
syl |
|- ( f : suc v -1-1-onto-> x -> ( f " suc v ) = x ) |
105 |
101 104
|
eqtrd |
|- ( f : suc v -1-1-onto-> x -> ( ( f " v ) u. { ( f ` v ) } ) = x ) |
106 |
105
|
inteqd |
|- ( f : suc v -1-1-onto-> x -> |^| ( ( f " v ) u. { ( f ` v ) } ) = |^| x ) |
107 |
92 106
|
eqtr3id |
|- ( f : suc v -1-1-onto-> x -> ( |^| ( f " v ) i^i ( f ` v ) ) = |^| x ) |
108 |
107
|
eleq1d |
|- ( f : suc v -1-1-onto-> x -> ( ( |^| ( f " v ) i^i ( f ` v ) ) e. A <-> |^| x e. A ) ) |
109 |
108
|
ad2antlr |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> ( ( |^| ( f " v ) i^i ( f ` v ) ) e. A <-> |^| x e. A ) ) |
110 |
90 109
|
mpbid |
|- ( ( ( x C_ A /\ f : suc v -1-1-onto-> x ) /\ ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) /\ A. z e. A A. w e. A ( z i^i w ) e. A ) ) -> |^| x e. A ) |
111 |
110
|
exp43 |
|- ( x C_ A -> ( f : suc v -1-1-onto-> x -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) ) |
112 |
111
|
exlimdv |
|- ( x C_ A -> ( E. f f : suc v -1-1-onto-> x -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) ) |
113 |
30 112
|
syl5bi |
|- ( x C_ A -> ( suc v ~~ x -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) ) |
114 |
113
|
imp |
|- ( ( x C_ A /\ suc v ~~ x ) -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) |
115 |
114
|
adantlr |
|- ( ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) |
116 |
115
|
com13 |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> ( ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) -> |^| x e. A ) ) ) |
117 |
28 29 116
|
alrimd |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) -> |^| x e. A ) ) ) |
118 |
117
|
a1i |
|- ( v e. _om -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ v ~~ x ) -> |^| x e. A ) -> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ suc v ~~ x ) -> |^| x e. A ) ) ) ) |
119 |
6 10 14 27 118
|
finds2 |
|- ( y e. _om -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) ) ) |
120 |
|
sp |
|- ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) -> ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) ) |
121 |
119 120
|
syl6 |
|- ( y e. _om -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( ( ( x C_ A /\ x =/= (/) ) /\ y ~~ x ) -> |^| x e. A ) ) ) |
122 |
121
|
exp4a |
|- ( y e. _om -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( ( x C_ A /\ x =/= (/) ) -> ( y ~~ x -> |^| x e. A ) ) ) ) |
123 |
122
|
com24 |
|- ( y e. _om -> ( y ~~ x -> ( ( x C_ A /\ x =/= (/) ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) ) |
124 |
2 123
|
syl5 |
|- ( y e. _om -> ( x ~~ y -> ( ( x C_ A /\ x =/= (/) ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) ) |
125 |
124
|
rexlimiv |
|- ( E. y e. _om x ~~ y -> ( ( x C_ A /\ x =/= (/) ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) |
126 |
1 125
|
sylbi |
|- ( x e. Fin -> ( ( x C_ A /\ x =/= (/) ) -> ( A. z e. A A. w e. A ( z i^i w ) e. A -> |^| x e. A ) ) ) |
127 |
126
|
com13 |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( ( x C_ A /\ x =/= (/) ) -> ( x e. Fin -> |^| x e. A ) ) ) |
128 |
127
|
impd |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) ) |
129 |
128
|
alrimiv |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A -> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) ) |
130 |
|
zfpair2 |
|- { z , w } e. _V |
131 |
|
sseq1 |
|- ( x = { z , w } -> ( x C_ A <-> { z , w } C_ A ) ) |
132 |
|
neeq1 |
|- ( x = { z , w } -> ( x =/= (/) <-> { z , w } =/= (/) ) ) |
133 |
131 132
|
anbi12d |
|- ( x = { z , w } -> ( ( x C_ A /\ x =/= (/) ) <-> ( { z , w } C_ A /\ { z , w } =/= (/) ) ) ) |
134 |
|
eleq1 |
|- ( x = { z , w } -> ( x e. Fin <-> { z , w } e. Fin ) ) |
135 |
133 134
|
anbi12d |
|- ( x = { z , w } -> ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) <-> ( ( { z , w } C_ A /\ { z , w } =/= (/) ) /\ { z , w } e. Fin ) ) ) |
136 |
|
inteq |
|- ( x = { z , w } -> |^| x = |^| { z , w } ) |
137 |
136
|
eleq1d |
|- ( x = { z , w } -> ( |^| x e. A <-> |^| { z , w } e. A ) ) |
138 |
135 137
|
imbi12d |
|- ( x = { z , w } -> ( ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) <-> ( ( ( { z , w } C_ A /\ { z , w } =/= (/) ) /\ { z , w } e. Fin ) -> |^| { z , w } e. A ) ) ) |
139 |
130 138
|
spcv |
|- ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) -> ( ( ( { z , w } C_ A /\ { z , w } =/= (/) ) /\ { z , w } e. Fin ) -> |^| { z , w } e. A ) ) |
140 |
|
vex |
|- z e. _V |
141 |
|
vex |
|- w e. _V |
142 |
140 141
|
prss |
|- ( ( z e. A /\ w e. A ) <-> { z , w } C_ A ) |
143 |
140
|
prnz |
|- { z , w } =/= (/) |
144 |
143
|
biantru |
|- ( { z , w } C_ A <-> ( { z , w } C_ A /\ { z , w } =/= (/) ) ) |
145 |
|
prfi |
|- { z , w } e. Fin |
146 |
145
|
biantru |
|- ( ( { z , w } C_ A /\ { z , w } =/= (/) ) <-> ( ( { z , w } C_ A /\ { z , w } =/= (/) ) /\ { z , w } e. Fin ) ) |
147 |
142 144 146
|
3bitrri |
|- ( ( ( { z , w } C_ A /\ { z , w } =/= (/) ) /\ { z , w } e. Fin ) <-> ( z e. A /\ w e. A ) ) |
148 |
140 141
|
intpr |
|- |^| { z , w } = ( z i^i w ) |
149 |
148
|
eleq1i |
|- ( |^| { z , w } e. A <-> ( z i^i w ) e. A ) |
150 |
139 147 149
|
3imtr3g |
|- ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) -> ( ( z e. A /\ w e. A ) -> ( z i^i w ) e. A ) ) |
151 |
150
|
ralrimivv |
|- ( A. x ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) -> A. z e. A A. w e. A ( z i^i w ) e. A ) |
152 |
129 151
|
impbii |
|- ( A. z e. A A. w e. A ( z i^i w ) e. A <-> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) ) |
153 |
|
ineq1 |
|- ( x = z -> ( x i^i y ) = ( z i^i y ) ) |
154 |
153
|
eleq1d |
|- ( x = z -> ( ( x i^i y ) e. A <-> ( z i^i y ) e. A ) ) |
155 |
|
ineq2 |
|- ( y = w -> ( z i^i y ) = ( z i^i w ) ) |
156 |
155
|
eleq1d |
|- ( y = w -> ( ( z i^i y ) e. A <-> ( z i^i w ) e. A ) ) |
157 |
154 156
|
cbvral2vw |
|- ( A. x e. A A. y e. A ( x i^i y ) e. A <-> A. z e. A A. w e. A ( z i^i w ) e. A ) |
158 |
|
df-3an |
|- ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) <-> ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) ) |
159 |
158
|
imbi1i |
|- ( ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) <-> ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) ) |
160 |
159
|
albii |
|- ( A. x ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) <-> A. x ( ( ( x C_ A /\ x =/= (/) ) /\ x e. Fin ) -> |^| x e. A ) ) |
161 |
152 157 160
|
3bitr4i |
|- ( A. x e. A A. y e. A ( x i^i y ) e. A <-> A. x ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) ) |