Step |
Hyp |
Ref |
Expression |
1 |
|
disjrnmpt2.1 |
|- F = ( x e. A |-> B ) |
2 |
|
id |
|- ( y = w -> y = w ) |
3 |
2
|
cbvdisjv |
|- ( Disj_ y e. ran F y <-> Disj_ w e. ran F w ) |
4 |
|
id |
|- ( w = v -> w = v ) |
5 |
4
|
ndisj2 |
|- ( -. Disj_ w e. ran F w <-> E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) ) |
6 |
5
|
biimpi |
|- ( -. Disj_ w e. ran F w -> E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) ) |
7 |
3 6
|
sylnbi |
|- ( -. Disj_ y e. ran F y -> E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) ) |
8 |
1
|
elrnmpt |
|- ( w e. ran F -> ( w e. ran F <-> E. x e. A w = B ) ) |
9 |
8
|
ibi |
|- ( w e. ran F -> E. x e. A w = B ) |
10 |
|
nfcv |
|- F/_ z B |
11 |
|
nfcsb1v |
|- F/_ x [_ z / x ]_ B |
12 |
|
csbeq1a |
|- ( x = z -> B = [_ z / x ]_ B ) |
13 |
10 11 12
|
cbvmpt |
|- ( x e. A |-> B ) = ( z e. A |-> [_ z / x ]_ B ) |
14 |
1 13
|
eqtri |
|- F = ( z e. A |-> [_ z / x ]_ B ) |
15 |
14
|
elrnmpt |
|- ( v e. ran F -> ( v e. ran F <-> E. z e. A v = [_ z / x ]_ B ) ) |
16 |
15
|
ibi |
|- ( v e. ran F -> E. z e. A v = [_ z / x ]_ B ) |
17 |
9 16
|
anim12i |
|- ( ( w e. ran F /\ v e. ran F ) -> ( E. x e. A w = B /\ E. z e. A v = [_ z / x ]_ B ) ) |
18 |
|
nfv |
|- F/ z w = B |
19 |
11
|
nfeq2 |
|- F/ x v = [_ z / x ]_ B |
20 |
18 19
|
reean |
|- ( E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) <-> ( E. x e. A w = B /\ E. z e. A v = [_ z / x ]_ B ) ) |
21 |
17 20
|
sylibr |
|- ( ( w e. ran F /\ v e. ran F ) -> E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) ) |
22 |
21
|
adantr |
|- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) ) |
23 |
|
nfmpt1 |
|- F/_ x ( x e. A |-> B ) |
24 |
1 23
|
nfcxfr |
|- F/_ x F |
25 |
24
|
nfrn |
|- F/_ x ran F |
26 |
25
|
nfcri |
|- F/ x w e. ran F |
27 |
25
|
nfcri |
|- F/ x v e. ran F |
28 |
26 27
|
nfan |
|- F/ x ( w e. ran F /\ v e. ran F ) |
29 |
|
nfv |
|- F/ x ( w =/= v /\ ( w i^i v ) =/= (/) ) |
30 |
28 29
|
nfan |
|- F/ x ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) |
31 |
|
simpll |
|- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> w = B ) |
32 |
12
|
adantl |
|- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> B = [_ z / x ]_ B ) |
33 |
|
id |
|- ( v = [_ z / x ]_ B -> v = [_ z / x ]_ B ) |
34 |
33
|
eqcomd |
|- ( v = [_ z / x ]_ B -> [_ z / x ]_ B = v ) |
35 |
34
|
ad2antlr |
|- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> [_ z / x ]_ B = v ) |
36 |
31 32 35
|
3eqtrd |
|- ( ( ( w = B /\ v = [_ z / x ]_ B ) /\ x = z ) -> w = v ) |
37 |
36
|
adantll |
|- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> w = v ) |
38 |
|
simpll |
|- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> w =/= v ) |
39 |
38
|
neneqd |
|- ( ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) /\ x = z ) -> -. w = v ) |
40 |
37 39
|
pm2.65da |
|- ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> -. x = z ) |
41 |
40
|
neqned |
|- ( ( w =/= v /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> x =/= z ) |
42 |
41
|
adantlr |
|- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> x =/= z ) |
43 |
|
id |
|- ( w = B -> w = B ) |
44 |
43
|
eqcomd |
|- ( w = B -> B = w ) |
45 |
44
|
ad2antrl |
|- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> B = w ) |
46 |
34
|
ad2antll |
|- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> [_ z / x ]_ B = v ) |
47 |
45 46
|
ineq12d |
|- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) = ( w i^i v ) ) |
48 |
|
simpl |
|- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( w i^i v ) =/= (/) ) |
49 |
47 48
|
eqnetrd |
|- ( ( ( w i^i v ) =/= (/) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) =/= (/) ) |
50 |
49
|
adantll |
|- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( B i^i [_ z / x ]_ B ) =/= (/) ) |
51 |
42 50
|
jca |
|- ( ( ( w =/= v /\ ( w i^i v ) =/= (/) ) /\ ( w = B /\ v = [_ z / x ]_ B ) ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
52 |
51
|
ex |
|- ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> ( ( w = B /\ v = [_ z / x ]_ B ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
53 |
52
|
adantl |
|- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( ( w = B /\ v = [_ z / x ]_ B ) -> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
54 |
53
|
reximdv |
|- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( E. z e. A ( w = B /\ v = [_ z / x ]_ B ) -> E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
55 |
54
|
a1d |
|- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( x e. A -> ( E. z e. A ( w = B /\ v = [_ z / x ]_ B ) -> E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) ) |
56 |
30 55
|
reximdai |
|- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> ( E. x e. A E. z e. A ( w = B /\ v = [_ z / x ]_ B ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
57 |
22 56
|
mpd |
|- ( ( ( w e. ran F /\ v e. ran F ) /\ ( w =/= v /\ ( w i^i v ) =/= (/) ) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
58 |
57
|
ex |
|- ( ( w e. ran F /\ v e. ran F ) -> ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
59 |
58
|
a1i |
|- ( -. Disj_ y e. ran F y -> ( ( w e. ran F /\ v e. ran F ) -> ( ( w =/= v /\ ( w i^i v ) =/= (/) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) ) |
60 |
59
|
rexlimdvv |
|- ( -. Disj_ y e. ran F y -> ( E. w e. ran F E. v e. ran F ( w =/= v /\ ( w i^i v ) =/= (/) ) -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
61 |
7 60
|
mpd |
|- ( -. Disj_ y e. ran F y -> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
62 |
|
csbeq1 |
|- ( u = z -> [_ u / x ]_ B = [_ z / x ]_ B ) |
63 |
62
|
ndisj2 |
|- ( -. Disj_ u e. A [_ u / x ]_ B <-> E. u e. A E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) ) |
64 |
|
nfcv |
|- F/_ x A |
65 |
|
nfv |
|- F/ x u =/= z |
66 |
|
nfcsb1v |
|- F/_ x [_ u / x ]_ B |
67 |
66 11
|
nfin |
|- F/_ x ( [_ u / x ]_ B i^i [_ z / x ]_ B ) |
68 |
|
nfcv |
|- F/_ x (/) |
69 |
67 68
|
nfne |
|- F/ x ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) |
70 |
65 69
|
nfan |
|- F/ x ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) |
71 |
64 70
|
nfrex |
|- F/ x E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) |
72 |
|
nfv |
|- F/ u E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) |
73 |
|
neeq1 |
|- ( u = x -> ( u =/= z <-> x =/= z ) ) |
74 |
|
csbeq1 |
|- ( u = x -> [_ u / x ]_ B = [_ x / x ]_ B ) |
75 |
|
csbid |
|- [_ x / x ]_ B = B |
76 |
74 75
|
eqtrdi |
|- ( u = x -> [_ u / x ]_ B = B ) |
77 |
76
|
ineq1d |
|- ( u = x -> ( [_ u / x ]_ B i^i [_ z / x ]_ B ) = ( B i^i [_ z / x ]_ B ) ) |
78 |
77
|
neeq1d |
|- ( u = x -> ( ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) <-> ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
79 |
73 78
|
anbi12d |
|- ( u = x -> ( ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
80 |
79
|
rexbidv |
|- ( u = x -> ( E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) ) |
81 |
71 72 80
|
cbvrexw |
|- ( E. u e. A E. z e. A ( u =/= z /\ ( [_ u / x ]_ B i^i [_ z / x ]_ B ) =/= (/) ) <-> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
82 |
63 81
|
bitri |
|- ( -. Disj_ u e. A [_ u / x ]_ B <-> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
83 |
|
nfcv |
|- F/_ u B |
84 |
|
csbeq1a |
|- ( x = u -> B = [_ u / x ]_ B ) |
85 |
83 66 84
|
cbvdisj |
|- ( Disj_ x e. A B <-> Disj_ u e. A [_ u / x ]_ B ) |
86 |
82 85
|
xchnxbir |
|- ( -. Disj_ x e. A B <-> E. x e. A E. z e. A ( x =/= z /\ ( B i^i [_ z / x ]_ B ) =/= (/) ) ) |
87 |
61 86
|
sylibr |
|- ( -. Disj_ y e. ran F y -> -. Disj_ x e. A B ) |
88 |
87
|
con4i |
|- ( Disj_ x e. A B -> Disj_ y e. ran F y ) |