Step |
Hyp |
Ref |
Expression |
1 |
|
brprop.a |
|- ( ph -> A e. V ) |
2 |
|
brprop.b |
|- ( ph -> B e. W ) |
3 |
|
brprop.c |
|- ( ph -> C e. V ) |
4 |
|
brprop.d |
|- ( ph -> D e. W ) |
5 |
|
mptprop.1 |
|- ( ph -> A =/= C ) |
6 |
|
df-pr |
|- { <. A , B >. , <. C , D >. } = ( { <. A , B >. } u. { <. C , D >. } ) |
7 |
|
fmptsn |
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } = ( x e. { A } |-> B ) ) |
8 |
1 2 7
|
syl2anc |
|- ( ph -> { <. A , B >. } = ( x e. { A } |-> B ) ) |
9 |
|
incom |
|- ( { A } i^i { A , C } ) = ( { A , C } i^i { A } ) |
10 |
|
prid1g |
|- ( A e. V -> A e. { A , C } ) |
11 |
|
snssi |
|- ( A e. { A , C } -> { A } C_ { A , C } ) |
12 |
1 10 11
|
3syl |
|- ( ph -> { A } C_ { A , C } ) |
13 |
|
df-ss |
|- ( { A } C_ { A , C } <-> ( { A } i^i { A , C } ) = { A } ) |
14 |
12 13
|
sylib |
|- ( ph -> ( { A } i^i { A , C } ) = { A } ) |
15 |
9 14
|
eqtr3id |
|- ( ph -> ( { A , C } i^i { A } ) = { A } ) |
16 |
15
|
mpteq1d |
|- ( ph -> ( x e. ( { A , C } i^i { A } ) |-> B ) = ( x e. { A } |-> B ) ) |
17 |
8 16
|
eqtr4d |
|- ( ph -> { <. A , B >. } = ( x e. ( { A , C } i^i { A } ) |-> B ) ) |
18 |
|
fmptsn |
|- ( ( C e. V /\ D e. W ) -> { <. C , D >. } = ( x e. { C } |-> D ) ) |
19 |
3 4 18
|
syl2anc |
|- ( ph -> { <. C , D >. } = ( x e. { C } |-> D ) ) |
20 |
|
difprsn1 |
|- ( A =/= C -> ( { A , C } \ { A } ) = { C } ) |
21 |
5 20
|
syl |
|- ( ph -> ( { A , C } \ { A } ) = { C } ) |
22 |
21
|
mpteq1d |
|- ( ph -> ( x e. ( { A , C } \ { A } ) |-> D ) = ( x e. { C } |-> D ) ) |
23 |
19 22
|
eqtr4d |
|- ( ph -> { <. C , D >. } = ( x e. ( { A , C } \ { A } ) |-> D ) ) |
24 |
17 23
|
uneq12d |
|- ( ph -> ( { <. A , B >. } u. { <. C , D >. } ) = ( ( x e. ( { A , C } i^i { A } ) |-> B ) u. ( x e. ( { A , C } \ { A } ) |-> D ) ) ) |
25 |
|
partfun |
|- ( x e. { A , C } |-> if ( x e. { A } , B , D ) ) = ( ( x e. ( { A , C } i^i { A } ) |-> B ) u. ( x e. ( { A , C } \ { A } ) |-> D ) ) |
26 |
24 25
|
eqtr4di |
|- ( ph -> ( { <. A , B >. } u. { <. C , D >. } ) = ( x e. { A , C } |-> if ( x e. { A } , B , D ) ) ) |
27 |
|
elsn2g |
|- ( A e. V -> ( x e. { A } <-> x = A ) ) |
28 |
1 27
|
syl |
|- ( ph -> ( x e. { A } <-> x = A ) ) |
29 |
28
|
ifbid |
|- ( ph -> if ( x e. { A } , B , D ) = if ( x = A , B , D ) ) |
30 |
29
|
mpteq2dv |
|- ( ph -> ( x e. { A , C } |-> if ( x e. { A } , B , D ) ) = ( x e. { A , C } |-> if ( x = A , B , D ) ) ) |
31 |
26 30
|
eqtrd |
|- ( ph -> ( { <. A , B >. } u. { <. C , D >. } ) = ( x e. { A , C } |-> if ( x = A , B , D ) ) ) |
32 |
6 31
|
syl5eq |
|- ( ph -> { <. A , B >. , <. C , D >. } = ( x e. { A , C } |-> if ( x = A , B , D ) ) ) |