Step |
Hyp |
Ref |
Expression |
1 |
|
xpcomf1o.1 |
|- F = ( x e. ( A X. B ) |-> U. `' { x } ) |
2 |
|
xpcomco.1 |
|- G = ( y e. B , z e. A |-> C ) |
3 |
1
|
xpcomf1o |
|- F : ( A X. B ) -1-1-onto-> ( B X. A ) |
4 |
|
f1ofun |
|- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> Fun F ) |
5 |
|
funbrfv2b |
|- ( Fun F -> ( u F w <-> ( u e. dom F /\ ( F ` u ) = w ) ) ) |
6 |
3 4 5
|
mp2b |
|- ( u F w <-> ( u e. dom F /\ ( F ` u ) = w ) ) |
7 |
|
ancom |
|- ( ( u e. dom F /\ ( F ` u ) = w ) <-> ( ( F ` u ) = w /\ u e. dom F ) ) |
8 |
|
eqcom |
|- ( ( F ` u ) = w <-> w = ( F ` u ) ) |
9 |
|
f1odm |
|- ( F : ( A X. B ) -1-1-onto-> ( B X. A ) -> dom F = ( A X. B ) ) |
10 |
3 9
|
ax-mp |
|- dom F = ( A X. B ) |
11 |
10
|
eleq2i |
|- ( u e. dom F <-> u e. ( A X. B ) ) |
12 |
8 11
|
anbi12i |
|- ( ( ( F ` u ) = w /\ u e. dom F ) <-> ( w = ( F ` u ) /\ u e. ( A X. B ) ) ) |
13 |
6 7 12
|
3bitri |
|- ( u F w <-> ( w = ( F ` u ) /\ u e. ( A X. B ) ) ) |
14 |
13
|
anbi1i |
|- ( ( u F w /\ w G v ) <-> ( ( w = ( F ` u ) /\ u e. ( A X. B ) ) /\ w G v ) ) |
15 |
|
anass |
|- ( ( ( w = ( F ` u ) /\ u e. ( A X. B ) ) /\ w G v ) <-> ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) ) |
16 |
14 15
|
bitri |
|- ( ( u F w /\ w G v ) <-> ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) ) |
17 |
16
|
exbii |
|- ( E. w ( u F w /\ w G v ) <-> E. w ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) ) |
18 |
|
fvex |
|- ( F ` u ) e. _V |
19 |
|
breq1 |
|- ( w = ( F ` u ) -> ( w G v <-> ( F ` u ) G v ) ) |
20 |
19
|
anbi2d |
|- ( w = ( F ` u ) -> ( ( u e. ( A X. B ) /\ w G v ) <-> ( u e. ( A X. B ) /\ ( F ` u ) G v ) ) ) |
21 |
18 20
|
ceqsexv |
|- ( E. w ( w = ( F ` u ) /\ ( u e. ( A X. B ) /\ w G v ) ) <-> ( u e. ( A X. B ) /\ ( F ` u ) G v ) ) |
22 |
|
elxp |
|- ( u e. ( A X. B ) <-> E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) ) |
23 |
22
|
anbi1i |
|- ( ( u e. ( A X. B ) /\ ( F ` u ) G v ) <-> ( E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) ) |
24 |
|
nfcv |
|- F/_ z ( F ` u ) |
25 |
|
nfmpo2 |
|- F/_ z ( y e. B , z e. A |-> C ) |
26 |
2 25
|
nfcxfr |
|- F/_ z G |
27 |
|
nfcv |
|- F/_ z v |
28 |
24 26 27
|
nfbr |
|- F/ z ( F ` u ) G v |
29 |
28
|
19.41 |
|- ( E. z ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( E. z E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) ) |
30 |
|
nfcv |
|- F/_ y ( F ` u ) |
31 |
|
nfmpo1 |
|- F/_ y ( y e. B , z e. A |-> C ) |
32 |
2 31
|
nfcxfr |
|- F/_ y G |
33 |
|
nfcv |
|- F/_ y v |
34 |
30 32 33
|
nfbr |
|- F/ y ( F ` u ) G v |
35 |
34
|
19.41 |
|- ( E. y ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) ) |
36 |
|
anass |
|- ( ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ ( F ` u ) G v ) ) ) |
37 |
|
fveq2 |
|- ( u = <. z , y >. -> ( F ` u ) = ( F ` <. z , y >. ) ) |
38 |
|
opelxpi |
|- ( ( z e. A /\ y e. B ) -> <. z , y >. e. ( A X. B ) ) |
39 |
|
sneq |
|- ( x = <. z , y >. -> { x } = { <. z , y >. } ) |
40 |
39
|
cnveqd |
|- ( x = <. z , y >. -> `' { x } = `' { <. z , y >. } ) |
41 |
40
|
unieqd |
|- ( x = <. z , y >. -> U. `' { x } = U. `' { <. z , y >. } ) |
42 |
|
opswap |
|- U. `' { <. z , y >. } = <. y , z >. |
43 |
41 42
|
eqtrdi |
|- ( x = <. z , y >. -> U. `' { x } = <. y , z >. ) |
44 |
|
opex |
|- <. y , z >. e. _V |
45 |
43 1 44
|
fvmpt |
|- ( <. z , y >. e. ( A X. B ) -> ( F ` <. z , y >. ) = <. y , z >. ) |
46 |
38 45
|
syl |
|- ( ( z e. A /\ y e. B ) -> ( F ` <. z , y >. ) = <. y , z >. ) |
47 |
37 46
|
sylan9eq |
|- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( F ` u ) = <. y , z >. ) |
48 |
47
|
breq1d |
|- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> <. y , z >. G v ) ) |
49 |
|
df-br |
|- ( <. y , z >. G v <-> <. <. y , z >. , v >. e. G ) |
50 |
|
df-mpo |
|- ( y e. B , z e. A |-> C ) = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } |
51 |
2 50
|
eqtri |
|- G = { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } |
52 |
51
|
eleq2i |
|- ( <. <. y , z >. , v >. e. G <-> <. <. y , z >. , v >. e. { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } ) |
53 |
|
oprabidw |
|- ( <. <. y , z >. , v >. e. { <. <. y , z >. , v >. | ( ( y e. B /\ z e. A ) /\ v = C ) } <-> ( ( y e. B /\ z e. A ) /\ v = C ) ) |
54 |
49 52 53
|
3bitri |
|- ( <. y , z >. G v <-> ( ( y e. B /\ z e. A ) /\ v = C ) ) |
55 |
54
|
baib |
|- ( ( y e. B /\ z e. A ) -> ( <. y , z >. G v <-> v = C ) ) |
56 |
55
|
ancoms |
|- ( ( z e. A /\ y e. B ) -> ( <. y , z >. G v <-> v = C ) ) |
57 |
56
|
adantl |
|- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( <. y , z >. G v <-> v = C ) ) |
58 |
48 57
|
bitrd |
|- ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) -> ( ( F ` u ) G v <-> v = C ) ) |
59 |
58
|
pm5.32da |
|- ( u = <. z , y >. -> ( ( ( z e. A /\ y e. B ) /\ ( F ` u ) G v ) <-> ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
60 |
59
|
pm5.32i |
|- ( ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ ( F ` u ) G v ) ) <-> ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
61 |
36 60
|
bitri |
|- ( ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
62 |
61
|
exbii |
|- ( E. y ( ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
63 |
35 62
|
bitr3i |
|- ( ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
64 |
63
|
exbii |
|- ( E. z ( E. y ( u = <. z , y >. /\ ( z e. A /\ y e. B ) ) /\ ( F ` u ) G v ) <-> E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
65 |
23 29 64
|
3bitr2i |
|- ( ( u e. ( A X. B ) /\ ( F ` u ) G v ) <-> E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
66 |
17 21 65
|
3bitri |
|- ( E. w ( u F w /\ w G v ) <-> E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) ) |
67 |
66
|
opabbii |
|- { <. u , v >. | E. w ( u F w /\ w G v ) } = { <. u , v >. | E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) } |
68 |
|
df-co |
|- ( G o. F ) = { <. u , v >. | E. w ( u F w /\ w G v ) } |
69 |
|
df-mpo |
|- ( z e. A , y e. B |-> C ) = { <. <. z , y >. , v >. | ( ( z e. A /\ y e. B ) /\ v = C ) } |
70 |
|
dfoprab2 |
|- { <. <. z , y >. , v >. | ( ( z e. A /\ y e. B ) /\ v = C ) } = { <. u , v >. | E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) } |
71 |
69 70
|
eqtri |
|- ( z e. A , y e. B |-> C ) = { <. u , v >. | E. z E. y ( u = <. z , y >. /\ ( ( z e. A /\ y e. B ) /\ v = C ) ) } |
72 |
67 68 71
|
3eqtr4i |
|- ( G o. F ) = ( z e. A , y e. B |-> C ) |