Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
2 |
|
psrbagconf1o.s |
|- S = { y e. D | y oR <_ F } |
3 |
|
eqid |
|- ( x e. S |-> ( F oF - x ) ) = ( x e. S |-> ( F oF - x ) ) |
4 |
1 2
|
psrbagconcl |
|- ( ( F e. D /\ x e. S ) -> ( F oF - x ) e. S ) |
5 |
1 2
|
psrbagconcl |
|- ( ( F e. D /\ z e. S ) -> ( F oF - z ) e. S ) |
6 |
1
|
psrbagf |
|- ( F e. D -> F : I --> NN0 ) |
7 |
6
|
adantr |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> F : I --> NN0 ) |
8 |
7
|
ffvelrnda |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) e. NN0 ) |
9 |
2
|
ssrab3 |
|- S C_ D |
10 |
9
|
sseli |
|- ( z e. S -> z e. D ) |
11 |
10
|
adantl |
|- ( ( F e. D /\ z e. S ) -> z e. D ) |
12 |
1
|
psrbagf |
|- ( z e. D -> z : I --> NN0 ) |
13 |
11 12
|
syl |
|- ( ( F e. D /\ z e. S ) -> z : I --> NN0 ) |
14 |
13
|
adantrl |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> z : I --> NN0 ) |
15 |
14
|
ffvelrnda |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) e. NN0 ) |
16 |
|
simprl |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x e. S ) |
17 |
9 16
|
sselid |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x e. D ) |
18 |
1
|
psrbagf |
|- ( x e. D -> x : I --> NN0 ) |
19 |
17 18
|
syl |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x : I --> NN0 ) |
20 |
19
|
ffvelrnda |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) e. NN0 ) |
21 |
|
nn0cn |
|- ( ( F ` n ) e. NN0 -> ( F ` n ) e. CC ) |
22 |
|
nn0cn |
|- ( ( z ` n ) e. NN0 -> ( z ` n ) e. CC ) |
23 |
|
nn0cn |
|- ( ( x ` n ) e. NN0 -> ( x ` n ) e. CC ) |
24 |
|
subsub23 |
|- ( ( ( F ` n ) e. CC /\ ( z ` n ) e. CC /\ ( x ` n ) e. CC ) -> ( ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) ) |
25 |
21 22 23 24
|
syl3an |
|- ( ( ( F ` n ) e. NN0 /\ ( z ` n ) e. NN0 /\ ( x ` n ) e. NN0 ) -> ( ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) ) |
26 |
8 15 20 25
|
syl3anc |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) ) |
27 |
|
eqcom |
|- ( ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) <-> ( ( F ` n ) - ( z ` n ) ) = ( x ` n ) ) |
28 |
|
eqcom |
|- ( ( z ` n ) = ( ( F ` n ) - ( x ` n ) ) <-> ( ( F ` n ) - ( x ` n ) ) = ( z ` n ) ) |
29 |
26 27 28
|
3bitr4g |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) <-> ( z ` n ) = ( ( F ` n ) - ( x ` n ) ) ) ) |
30 |
6
|
ffnd |
|- ( F e. D -> F Fn I ) |
31 |
30
|
adantr |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> F Fn I ) |
32 |
13
|
ffnd |
|- ( ( F e. D /\ z e. S ) -> z Fn I ) |
33 |
32
|
adantrl |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> z Fn I ) |
34 |
19
|
ffnd |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> x Fn I ) |
35 |
16 34
|
fndmexd |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> I e. _V ) |
36 |
|
inidm |
|- ( I i^i I ) = I |
37 |
|
eqidd |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( F ` n ) = ( F ` n ) ) |
38 |
|
eqidd |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( z ` n ) = ( z ` n ) ) |
39 |
31 33 35 35 36 37 38
|
ofval |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F oF - z ) ` n ) = ( ( F ` n ) - ( z ` n ) ) ) |
40 |
39
|
eqeq2d |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( x ` n ) = ( ( F oF - z ) ` n ) <-> ( x ` n ) = ( ( F ` n ) - ( z ` n ) ) ) ) |
41 |
|
eqidd |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( x ` n ) = ( x ` n ) ) |
42 |
31 34 35 35 36 37 41
|
ofval |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( F oF - x ) ` n ) = ( ( F ` n ) - ( x ` n ) ) ) |
43 |
42
|
eqeq2d |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( z ` n ) = ( ( F oF - x ) ` n ) <-> ( z ` n ) = ( ( F ` n ) - ( x ` n ) ) ) ) |
44 |
29 40 43
|
3bitr4d |
|- ( ( ( F e. D /\ ( x e. S /\ z e. S ) ) /\ n e. I ) -> ( ( x ` n ) = ( ( F oF - z ) ` n ) <-> ( z ` n ) = ( ( F oF - x ) ` n ) ) ) |
45 |
44
|
ralbidva |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( A. n e. I ( x ` n ) = ( ( F oF - z ) ` n ) <-> A. n e. I ( z ` n ) = ( ( F oF - x ) ` n ) ) ) |
46 |
5
|
adantrl |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) e. S ) |
47 |
9 46
|
sselid |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) e. D ) |
48 |
1
|
psrbagf |
|- ( ( F oF - z ) e. D -> ( F oF - z ) : I --> NN0 ) |
49 |
47 48
|
syl |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) : I --> NN0 ) |
50 |
49
|
ffnd |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - z ) Fn I ) |
51 |
|
eqfnfv |
|- ( ( x Fn I /\ ( F oF - z ) Fn I ) -> ( x = ( F oF - z ) <-> A. n e. I ( x ` n ) = ( ( F oF - z ) ` n ) ) ) |
52 |
34 50 51
|
syl2anc |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( x = ( F oF - z ) <-> A. n e. I ( x ` n ) = ( ( F oF - z ) ` n ) ) ) |
53 |
9 4
|
sselid |
|- ( ( F e. D /\ x e. S ) -> ( F oF - x ) e. D ) |
54 |
1
|
psrbagf |
|- ( ( F oF - x ) e. D -> ( F oF - x ) : I --> NN0 ) |
55 |
53 54
|
syl |
|- ( ( F e. D /\ x e. S ) -> ( F oF - x ) : I --> NN0 ) |
56 |
55
|
ffnd |
|- ( ( F e. D /\ x e. S ) -> ( F oF - x ) Fn I ) |
57 |
56
|
adantrr |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( F oF - x ) Fn I ) |
58 |
|
eqfnfv |
|- ( ( z Fn I /\ ( F oF - x ) Fn I ) -> ( z = ( F oF - x ) <-> A. n e. I ( z ` n ) = ( ( F oF - x ) ` n ) ) ) |
59 |
33 57 58
|
syl2anc |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( z = ( F oF - x ) <-> A. n e. I ( z ` n ) = ( ( F oF - x ) ` n ) ) ) |
60 |
45 52 59
|
3bitr4d |
|- ( ( F e. D /\ ( x e. S /\ z e. S ) ) -> ( x = ( F oF - z ) <-> z = ( F oF - x ) ) ) |
61 |
3 4 5 60
|
f1o2d |
|- ( F e. D -> ( x e. S |-> ( F oF - x ) ) : S -1-1-onto-> S ) |