Step |
Hyp |
Ref |
Expression |
1 |
|
brapply.1 |
|- A e. _V |
2 |
|
brapply.2 |
|- B e. _V |
3 |
|
brapply.3 |
|- C e. _V |
4 |
|
snex |
|- { ( A " { B } ) } e. _V |
5 |
4
|
inex1 |
|- ( { ( A " { B } ) } i^i Singletons ) e. _V |
6 |
|
unieq |
|- ( x = ( { ( A " { B } ) } i^i Singletons ) -> U. x = U. ( { ( A " { B } ) } i^i Singletons ) ) |
7 |
6
|
unieqd |
|- ( x = ( { ( A " { B } ) } i^i Singletons ) -> U. U. x = U. U. ( { ( A " { B } ) } i^i Singletons ) ) |
8 |
7
|
eqeq2d |
|- ( x = ( { ( A " { B } ) } i^i Singletons ) -> ( C = U. U. x <-> C = U. U. ( { ( A " { B } ) } i^i Singletons ) ) ) |
9 |
5 8
|
ceqsexv |
|- ( E. x ( x = ( { ( A " { B } ) } i^i Singletons ) /\ C = U. U. x ) <-> C = U. U. ( { ( A " { B } ) } i^i Singletons ) ) |
10 |
|
df-apply |
|- Apply = ( ( Bigcup o. Bigcup ) o. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) ) |
11 |
10
|
breqi |
|- ( <. A , B >. Apply C <-> <. A , B >. ( ( Bigcup o. Bigcup ) o. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) ) C ) |
12 |
|
opex |
|- <. A , B >. e. _V |
13 |
12 3
|
brco |
|- ( <. A , B >. ( ( Bigcup o. Bigcup ) o. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) ) C <-> E. x ( <. A , B >. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) x /\ x ( Bigcup o. Bigcup ) C ) ) |
14 |
|
vex |
|- x e. _V |
15 |
12 14
|
brco |
|- ( <. A , B >. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) x <-> E. y ( <. A , B >. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) y /\ y ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) x ) ) |
16 |
|
vex |
|- y e. _V |
17 |
12 16
|
brco |
|- ( <. A , B >. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) y <-> E. z ( <. A , B >. pprod ( _I , Singleton ) z /\ z ( Singleton o. Img ) y ) ) |
18 |
|
vex |
|- z e. _V |
19 |
1 2 18
|
brpprod3a |
|- ( <. A , B >. pprod ( _I , Singleton ) z <-> E. a E. b ( z = <. a , b >. /\ A _I a /\ B Singleton b ) ) |
20 |
|
3anrot |
|- ( ( z = <. a , b >. /\ A _I a /\ B Singleton b ) <-> ( A _I a /\ B Singleton b /\ z = <. a , b >. ) ) |
21 |
|
vex |
|- a e. _V |
22 |
21
|
ideq |
|- ( A _I a <-> A = a ) |
23 |
|
eqcom |
|- ( A = a <-> a = A ) |
24 |
22 23
|
bitri |
|- ( A _I a <-> a = A ) |
25 |
|
vex |
|- b e. _V |
26 |
2 25
|
brsingle |
|- ( B Singleton b <-> b = { B } ) |
27 |
|
biid |
|- ( z = <. a , b >. <-> z = <. a , b >. ) |
28 |
24 26 27
|
3anbi123i |
|- ( ( A _I a /\ B Singleton b /\ z = <. a , b >. ) <-> ( a = A /\ b = { B } /\ z = <. a , b >. ) ) |
29 |
20 28
|
bitri |
|- ( ( z = <. a , b >. /\ A _I a /\ B Singleton b ) <-> ( a = A /\ b = { B } /\ z = <. a , b >. ) ) |
30 |
29
|
2exbii |
|- ( E. a E. b ( z = <. a , b >. /\ A _I a /\ B Singleton b ) <-> E. a E. b ( a = A /\ b = { B } /\ z = <. a , b >. ) ) |
31 |
|
snex |
|- { B } e. _V |
32 |
|
opeq1 |
|- ( a = A -> <. a , b >. = <. A , b >. ) |
33 |
32
|
eqeq2d |
|- ( a = A -> ( z = <. a , b >. <-> z = <. A , b >. ) ) |
34 |
|
opeq2 |
|- ( b = { B } -> <. A , b >. = <. A , { B } >. ) |
35 |
34
|
eqeq2d |
|- ( b = { B } -> ( z = <. A , b >. <-> z = <. A , { B } >. ) ) |
36 |
1 31 33 35
|
ceqsex2v |
|- ( E. a E. b ( a = A /\ b = { B } /\ z = <. a , b >. ) <-> z = <. A , { B } >. ) |
37 |
19 30 36
|
3bitri |
|- ( <. A , B >. pprod ( _I , Singleton ) z <-> z = <. A , { B } >. ) |
38 |
37
|
anbi1i |
|- ( ( <. A , B >. pprod ( _I , Singleton ) z /\ z ( Singleton o. Img ) y ) <-> ( z = <. A , { B } >. /\ z ( Singleton o. Img ) y ) ) |
39 |
38
|
exbii |
|- ( E. z ( <. A , B >. pprod ( _I , Singleton ) z /\ z ( Singleton o. Img ) y ) <-> E. z ( z = <. A , { B } >. /\ z ( Singleton o. Img ) y ) ) |
40 |
|
opex |
|- <. A , { B } >. e. _V |
41 |
|
breq1 |
|- ( z = <. A , { B } >. -> ( z ( Singleton o. Img ) y <-> <. A , { B } >. ( Singleton o. Img ) y ) ) |
42 |
40 41
|
ceqsexv |
|- ( E. z ( z = <. A , { B } >. /\ z ( Singleton o. Img ) y ) <-> <. A , { B } >. ( Singleton o. Img ) y ) |
43 |
40 16
|
brco |
|- ( <. A , { B } >. ( Singleton o. Img ) y <-> E. x ( <. A , { B } >. Img x /\ x Singleton y ) ) |
44 |
1 31 14
|
brimg |
|- ( <. A , { B } >. Img x <-> x = ( A " { B } ) ) |
45 |
14 16
|
brsingle |
|- ( x Singleton y <-> y = { x } ) |
46 |
44 45
|
anbi12i |
|- ( ( <. A , { B } >. Img x /\ x Singleton y ) <-> ( x = ( A " { B } ) /\ y = { x } ) ) |
47 |
46
|
exbii |
|- ( E. x ( <. A , { B } >. Img x /\ x Singleton y ) <-> E. x ( x = ( A " { B } ) /\ y = { x } ) ) |
48 |
1
|
imaex |
|- ( A " { B } ) e. _V |
49 |
|
sneq |
|- ( x = ( A " { B } ) -> { x } = { ( A " { B } ) } ) |
50 |
49
|
eqeq2d |
|- ( x = ( A " { B } ) -> ( y = { x } <-> y = { ( A " { B } ) } ) ) |
51 |
48 50
|
ceqsexv |
|- ( E. x ( x = ( A " { B } ) /\ y = { x } ) <-> y = { ( A " { B } ) } ) |
52 |
47 51
|
bitri |
|- ( E. x ( <. A , { B } >. Img x /\ x Singleton y ) <-> y = { ( A " { B } ) } ) |
53 |
42 43 52
|
3bitri |
|- ( E. z ( z = <. A , { B } >. /\ z ( Singleton o. Img ) y ) <-> y = { ( A " { B } ) } ) |
54 |
17 39 53
|
3bitri |
|- ( <. A , B >. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) y <-> y = { ( A " { B } ) } ) |
55 |
|
eqid |
|- ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) |
56 |
|
brxp |
|- ( y ( _V X. _V ) x <-> ( y e. _V /\ x e. _V ) ) |
57 |
16 14 56
|
mpbir2an |
|- y ( _V X. _V ) x |
58 |
|
epel |
|- ( z _E y <-> z e. y ) |
59 |
58
|
anbi1ci |
|- ( ( z e. Singletons /\ z _E y ) <-> ( z e. y /\ z e. Singletons ) ) |
60 |
16
|
brresi |
|- ( z ( _E |` Singletons ) y <-> ( z e. Singletons /\ z _E y ) ) |
61 |
|
elin |
|- ( z e. ( y i^i Singletons ) <-> ( z e. y /\ z e. Singletons ) ) |
62 |
59 60 61
|
3bitr4ri |
|- ( z e. ( y i^i Singletons ) <-> z ( _E |` Singletons ) y ) |
63 |
16 14 55 57 62
|
brtxpsd3 |
|- ( y ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) x <-> x = ( y i^i Singletons ) ) |
64 |
54 63
|
anbi12i |
|- ( ( <. A , B >. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) y /\ y ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) x ) <-> ( y = { ( A " { B } ) } /\ x = ( y i^i Singletons ) ) ) |
65 |
64
|
exbii |
|- ( E. y ( <. A , B >. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) y /\ y ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) x ) <-> E. y ( y = { ( A " { B } ) } /\ x = ( y i^i Singletons ) ) ) |
66 |
|
ineq1 |
|- ( y = { ( A " { B } ) } -> ( y i^i Singletons ) = ( { ( A " { B } ) } i^i Singletons ) ) |
67 |
66
|
eqeq2d |
|- ( y = { ( A " { B } ) } -> ( x = ( y i^i Singletons ) <-> x = ( { ( A " { B } ) } i^i Singletons ) ) ) |
68 |
4 67
|
ceqsexv |
|- ( E. y ( y = { ( A " { B } ) } /\ x = ( y i^i Singletons ) ) <-> x = ( { ( A " { B } ) } i^i Singletons ) ) |
69 |
15 65 68
|
3bitri |
|- ( <. A , B >. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) x <-> x = ( { ( A " { B } ) } i^i Singletons ) ) |
70 |
14 3
|
brco |
|- ( x ( Bigcup o. Bigcup ) C <-> E. y ( x Bigcup y /\ y Bigcup C ) ) |
71 |
16
|
brbigcup |
|- ( x Bigcup y <-> U. x = y ) |
72 |
|
eqcom |
|- ( U. x = y <-> y = U. x ) |
73 |
71 72
|
bitri |
|- ( x Bigcup y <-> y = U. x ) |
74 |
3
|
brbigcup |
|- ( y Bigcup C <-> U. y = C ) |
75 |
|
eqcom |
|- ( U. y = C <-> C = U. y ) |
76 |
74 75
|
bitri |
|- ( y Bigcup C <-> C = U. y ) |
77 |
73 76
|
anbi12i |
|- ( ( x Bigcup y /\ y Bigcup C ) <-> ( y = U. x /\ C = U. y ) ) |
78 |
77
|
exbii |
|- ( E. y ( x Bigcup y /\ y Bigcup C ) <-> E. y ( y = U. x /\ C = U. y ) ) |
79 |
|
vuniex |
|- U. x e. _V |
80 |
|
unieq |
|- ( y = U. x -> U. y = U. U. x ) |
81 |
80
|
eqeq2d |
|- ( y = U. x -> ( C = U. y <-> C = U. U. x ) ) |
82 |
79 81
|
ceqsexv |
|- ( E. y ( y = U. x /\ C = U. y ) <-> C = U. U. x ) |
83 |
70 78 82
|
3bitri |
|- ( x ( Bigcup o. Bigcup ) C <-> C = U. U. x ) |
84 |
69 83
|
anbi12i |
|- ( ( <. A , B >. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) x /\ x ( Bigcup o. Bigcup ) C ) <-> ( x = ( { ( A " { B } ) } i^i Singletons ) /\ C = U. U. x ) ) |
85 |
84
|
exbii |
|- ( E. x ( <. A , B >. ( ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E |` Singletons ) (x) _V ) ) ) o. ( ( Singleton o. Img ) o. pprod ( _I , Singleton ) ) ) x /\ x ( Bigcup o. Bigcup ) C ) <-> E. x ( x = ( { ( A " { B } ) } i^i Singletons ) /\ C = U. U. x ) ) |
86 |
11 13 85
|
3bitri |
|- ( <. A , B >. Apply C <-> E. x ( x = ( { ( A " { B } ) } i^i Singletons ) /\ C = U. U. x ) ) |
87 |
|
dffv5 |
|- ( A ` B ) = U. U. ( { ( A " { B } ) } i^i Singletons ) |
88 |
87
|
eqeq2i |
|- ( C = ( A ` B ) <-> C = U. U. ( { ( A " { B } ) } i^i Singletons ) ) |
89 |
9 86 88
|
3bitr4i |
|- ( <. A , B >. Apply C <-> C = ( A ` B ) ) |