Step |
Hyp |
Ref |
Expression |
1 |
|
elxp7 |
|- ( w e. ( B X. C ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) ) |
2 |
1
|
anbi2i |
|- ( ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) <-> ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) ) ) |
3 |
|
anass |
|- ( ( ( ( 1st ` w ) e. A /\ ( 1st ` w ) e. B ) /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) <-> ( ( 1st ` w ) e. A /\ ( ( 1st ` w ) e. B /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) ) |
4 |
3
|
a1i |
|- ( A C_ B -> ( ( ( ( 1st ` w ) e. A /\ ( 1st ` w ) e. B ) /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) <-> ( ( 1st ` w ) e. A /\ ( ( 1st ` w ) e. B /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) ) ) |
5 |
|
ssel |
|- ( A C_ B -> ( ( 1st ` w ) e. A -> ( 1st ` w ) e. B ) ) |
6 |
5
|
pm4.71d |
|- ( A C_ B -> ( ( 1st ` w ) e. A <-> ( ( 1st ` w ) e. A /\ ( 1st ` w ) e. B ) ) ) |
7 |
6
|
anbi1d |
|- ( A C_ B -> ( ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) <-> ( ( ( 1st ` w ) e. A /\ ( 1st ` w ) e. B ) /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) ) |
8 |
|
an12 |
|- ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) <-> ( ( 1st ` w ) e. B /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) |
9 |
8
|
anbi2i |
|- ( ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) ) <-> ( ( 1st ` w ) e. A /\ ( ( 1st ` w ) e. B /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) ) |
10 |
9
|
a1i |
|- ( A C_ B -> ( ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) ) <-> ( ( 1st ` w ) e. A /\ ( ( 1st ` w ) e. B /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) ) ) |
11 |
4 7 10
|
3bitr4d |
|- ( A C_ B -> ( ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) <-> ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) ) ) ) |
12 |
2 11
|
bitr4id |
|- ( A C_ B -> ( ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) <-> ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) ) ) |
13 |
|
an12 |
|- ( ( ( 1st ` w ) e. A /\ ( w e. ( _V X. _V ) /\ ( 2nd ` w ) e. C ) ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. C ) ) ) |
14 |
12 13
|
bitrdi |
|- ( A C_ B -> ( ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. C ) ) ) ) |
15 |
|
cnvresima |
|- ( `' ( 1st |` ( B X. C ) ) " A ) = ( ( `' 1st " A ) i^i ( B X. C ) ) |
16 |
15
|
eleq2i |
|- ( w e. ( `' ( 1st |` ( B X. C ) ) " A ) <-> w e. ( ( `' 1st " A ) i^i ( B X. C ) ) ) |
17 |
|
elin |
|- ( w e. ( ( `' 1st " A ) i^i ( B X. C ) ) <-> ( w e. ( `' 1st " A ) /\ w e. ( B X. C ) ) ) |
18 |
|
vex |
|- w e. _V |
19 |
|
fo1st |
|- 1st : _V -onto-> _V |
20 |
|
fofn |
|- ( 1st : _V -onto-> _V -> 1st Fn _V ) |
21 |
|
elpreima |
|- ( 1st Fn _V -> ( w e. ( `' 1st " A ) <-> ( w e. _V /\ ( 1st ` w ) e. A ) ) ) |
22 |
19 20 21
|
mp2b |
|- ( w e. ( `' 1st " A ) <-> ( w e. _V /\ ( 1st ` w ) e. A ) ) |
23 |
18 22
|
mpbiran |
|- ( w e. ( `' 1st " A ) <-> ( 1st ` w ) e. A ) |
24 |
23
|
anbi1i |
|- ( ( w e. ( `' 1st " A ) /\ w e. ( B X. C ) ) <-> ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) ) |
25 |
16 17 24
|
3bitri |
|- ( w e. ( `' ( 1st |` ( B X. C ) ) " A ) <-> ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) ) |
26 |
|
elxp7 |
|- ( w e. ( A X. C ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. C ) ) ) |
27 |
14 25 26
|
3bitr4g |
|- ( A C_ B -> ( w e. ( `' ( 1st |` ( B X. C ) ) " A ) <-> w e. ( A X. C ) ) ) |
28 |
27
|
eqrdv |
|- ( A C_ B -> ( `' ( 1st |` ( B X. C ) ) " A ) = ( A X. C ) ) |