Step |
Hyp |
Ref |
Expression |
1 |
|
opabex3.1 |
|- A e. _V |
2 |
|
opabex3.2 |
|- ( x e. A -> { y | ph } e. _V ) |
3 |
|
19.42v |
|- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ph ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ph ) ) ) |
4 |
|
an12 |
|- ( ( z = <. x , y >. /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ph ) ) ) |
5 |
4
|
exbii |
|- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ph ) ) ) |
6 |
|
elxp |
|- ( z e. ( { x } X. { y | ph } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) ) |
7 |
|
excom |
|- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) ) |
8 |
|
an12 |
|- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) ) |
9 |
|
velsn |
|- ( v e. { x } <-> v = x ) |
10 |
9
|
anbi1i |
|- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) ) |
11 |
8 10
|
bitri |
|- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) ) |
12 |
11
|
exbii |
|- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) ) |
13 |
|
opeq1 |
|- ( v = x -> <. v , w >. = <. x , w >. ) |
14 |
13
|
eqeq2d |
|- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) ) |
15 |
14
|
anbi1d |
|- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ph } ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) ) ) |
16 |
15
|
equsexvw |
|- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) ) |
17 |
12 16
|
bitri |
|- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) ) |
18 |
17
|
exbii |
|- ( E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ph } ) ) |
19 |
7 18
|
bitri |
|- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ph } ) ) |
20 |
|
nfv |
|- F/ y z = <. x , w >. |
21 |
|
nfsab1 |
|- F/ y w e. { y | ph } |
22 |
20 21
|
nfan |
|- F/ y ( z = <. x , w >. /\ w e. { y | ph } ) |
23 |
|
nfv |
|- F/ w ( z = <. x , y >. /\ ph ) |
24 |
|
opeq2 |
|- ( w = y -> <. x , w >. = <. x , y >. ) |
25 |
24
|
eqeq2d |
|- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) ) |
26 |
|
df-clab |
|- ( w e. { y | ph } <-> [ w / y ] ph ) |
27 |
|
sbequ12 |
|- ( y = w -> ( ph <-> [ w / y ] ph ) ) |
28 |
27
|
equcoms |
|- ( w = y -> ( ph <-> [ w / y ] ph ) ) |
29 |
26 28
|
bitr4id |
|- ( w = y -> ( w e. { y | ph } <-> ph ) ) |
30 |
25 29
|
anbi12d |
|- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ph } ) <-> ( z = <. x , y >. /\ ph ) ) ) |
31 |
22 23 30
|
cbvexv1 |
|- ( E. w ( z = <. x , w >. /\ w e. { y | ph } ) <-> E. y ( z = <. x , y >. /\ ph ) ) |
32 |
6 19 31
|
3bitri |
|- ( z e. ( { x } X. { y | ph } ) <-> E. y ( z = <. x , y >. /\ ph ) ) |
33 |
32
|
anbi2i |
|- ( ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ph ) ) ) |
34 |
3 5 33
|
3bitr4ri |
|- ( ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) ) |
35 |
34
|
exbii |
|- ( E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) ) |
36 |
|
eliun |
|- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> E. x e. A z e. ( { x } X. { y | ph } ) ) |
37 |
|
df-rex |
|- ( E. x e. A z e. ( { x } X. { y | ph } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) ) |
38 |
36 37
|
bitri |
|- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) ) |
39 |
|
elopab |
|- ( z e. { <. x , y >. | ( x e. A /\ ph ) } <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) ) |
40 |
35 38 39
|
3bitr4i |
|- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> z e. { <. x , y >. | ( x e. A /\ ph ) } ) |
41 |
40
|
eqriv |
|- U_ x e. A ( { x } X. { y | ph } ) = { <. x , y >. | ( x e. A /\ ph ) } |
42 |
|
snex |
|- { x } e. _V |
43 |
|
xpexg |
|- ( ( { x } e. _V /\ { y | ph } e. _V ) -> ( { x } X. { y | ph } ) e. _V ) |
44 |
42 2 43
|
sylancr |
|- ( x e. A -> ( { x } X. { y | ph } ) e. _V ) |
45 |
44
|
rgen |
|- A. x e. A ( { x } X. { y | ph } ) e. _V |
46 |
|
iunexg |
|- ( ( A e. _V /\ A. x e. A ( { x } X. { y | ph } ) e. _V ) -> U_ x e. A ( { x } X. { y | ph } ) e. _V ) |
47 |
1 45 46
|
mp2an |
|- U_ x e. A ( { x } X. { y | ph } ) e. _V |
48 |
41 47
|
eqeltrri |
|- { <. x , y >. | ( x e. A /\ ph ) } e. _V |