Step |
Hyp |
Ref |
Expression |
1 |
|
bj-brresdm |
|- ( A ( _I |` C ) B -> A e. C ) |
2 |
|
relres |
|- Rel ( _I |` C ) |
3 |
2
|
brrelex2i |
|- ( A ( _I |` C ) B -> B e. _V ) |
4 |
1 3
|
jca |
|- ( A ( _I |` C ) B -> ( A e. C /\ B e. _V ) ) |
5 |
4
|
adantl |
|- ( ( ( A i^i B ) e. C /\ A ( _I |` C ) B ) -> ( A e. C /\ B e. _V ) ) |
6 |
|
eqimss |
|- ( A = B -> A C_ B ) |
7 |
|
df-ss |
|- ( A C_ B <-> ( A i^i B ) = A ) |
8 |
6 7
|
sylib |
|- ( A = B -> ( A i^i B ) = A ) |
9 |
8
|
adantl |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> ( A i^i B ) = A ) |
10 |
|
simpl |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> ( A i^i B ) e. C ) |
11 |
9 10
|
eqeltrrd |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> A e. C ) |
12 |
|
eqimss2 |
|- ( A = B -> B C_ A ) |
13 |
|
sseqin2 |
|- ( B C_ A <-> ( A i^i B ) = B ) |
14 |
12 13
|
sylib |
|- ( A = B -> ( A i^i B ) = B ) |
15 |
14
|
adantl |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> ( A i^i B ) = B ) |
16 |
15 10
|
eqeltrrd |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> B e. C ) |
17 |
16
|
elexd |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> B e. _V ) |
18 |
11 17
|
jca |
|- ( ( ( A i^i B ) e. C /\ A = B ) -> ( A e. C /\ B e. _V ) ) |
19 |
|
brres |
|- ( B e. _V -> ( A ( _I |` C ) B <-> ( A e. C /\ A _I B ) ) ) |
20 |
19
|
adantl |
|- ( ( A e. C /\ B e. _V ) -> ( A ( _I |` C ) B <-> ( A e. C /\ A _I B ) ) ) |
21 |
|
eqeq12 |
|- ( ( x = A /\ y = B ) -> ( x = y <-> A = B ) ) |
22 |
|
df-id |
|- _I = { <. x , y >. | x = y } |
23 |
21 22
|
brabga |
|- ( ( A e. C /\ B e. _V ) -> ( A _I B <-> A = B ) ) |
24 |
23
|
anbi2d |
|- ( ( A e. C /\ B e. _V ) -> ( ( A e. C /\ A _I B ) <-> ( A e. C /\ A = B ) ) ) |
25 |
|
simp3 |
|- ( ( ( A e. C /\ B e. _V ) /\ A e. C /\ A = B ) -> A = B ) |
26 |
25
|
3expib |
|- ( ( A e. C /\ B e. _V ) -> ( ( A e. C /\ A = B ) -> A = B ) ) |
27 |
|
3simpb |
|- ( ( A e. C /\ B e. _V /\ A = B ) -> ( A e. C /\ A = B ) ) |
28 |
27
|
3expia |
|- ( ( A e. C /\ B e. _V ) -> ( A = B -> ( A e. C /\ A = B ) ) ) |
29 |
26 28
|
impbid |
|- ( ( A e. C /\ B e. _V ) -> ( ( A e. C /\ A = B ) <-> A = B ) ) |
30 |
20 24 29
|
3bitrd |
|- ( ( A e. C /\ B e. _V ) -> ( A ( _I |` C ) B <-> A = B ) ) |
31 |
5 18 30
|
pm5.21nd |
|- ( ( A i^i B ) e. C -> ( A ( _I |` C ) B <-> A = B ) ) |