Step |
Hyp |
Ref |
Expression |
1 |
|
ecopopr.1 |
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) } |
2 |
|
ecopopr.com |
|- ( x .+ y ) = ( y .+ x ) |
3 |
|
ecopopr.cl |
|- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) |
4 |
|
ecopopr.ass |
|- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) |
5 |
|
ecopopr.can |
|- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) ) |
6 |
1
|
relopabiv |
|- Rel .~ |
7 |
1 2
|
ecopovsym |
|- ( f .~ g -> g .~ f ) |
8 |
1 2 3 4 5
|
ecopovtrn |
|- ( ( f .~ g /\ g .~ h ) -> f .~ h ) |
9 |
|
vex |
|- g e. _V |
10 |
|
vex |
|- h e. _V |
11 |
9 10 2
|
caovcom |
|- ( g .+ h ) = ( h .+ g ) |
12 |
1
|
ecopoveq |
|- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( <. g , h >. .~ <. g , h >. <-> ( g .+ h ) = ( h .+ g ) ) ) |
13 |
11 12
|
mpbiri |
|- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> <. g , h >. .~ <. g , h >. ) |
14 |
13
|
anidms |
|- ( ( g e. S /\ h e. S ) -> <. g , h >. .~ <. g , h >. ) |
15 |
14
|
rgen2 |
|- A. g e. S A. h e. S <. g , h >. .~ <. g , h >. |
16 |
|
breq12 |
|- ( ( f = <. g , h >. /\ f = <. g , h >. ) -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) ) |
17 |
16
|
anidms |
|- ( f = <. g , h >. -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) ) |
18 |
17
|
ralxp |
|- ( A. f e. ( S X. S ) f .~ f <-> A. g e. S A. h e. S <. g , h >. .~ <. g , h >. ) |
19 |
15 18
|
mpbir |
|- A. f e. ( S X. S ) f .~ f |
20 |
19
|
rspec |
|- ( f e. ( S X. S ) -> f .~ f ) |
21 |
|
opabssxp |
|- { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) } C_ ( ( S X. S ) X. ( S X. S ) ) |
22 |
1 21
|
eqsstri |
|- .~ C_ ( ( S X. S ) X. ( S X. S ) ) |
23 |
22
|
ssbri |
|- ( f .~ f -> f ( ( S X. S ) X. ( S X. S ) ) f ) |
24 |
|
brxp |
|- ( f ( ( S X. S ) X. ( S X. S ) ) f <-> ( f e. ( S X. S ) /\ f e. ( S X. S ) ) ) |
25 |
24
|
simplbi |
|- ( f ( ( S X. S ) X. ( S X. S ) ) f -> f e. ( S X. S ) ) |
26 |
23 25
|
syl |
|- ( f .~ f -> f e. ( S X. S ) ) |
27 |
20 26
|
impbii |
|- ( f e. ( S X. S ) <-> f .~ f ) |
28 |
6 7 8 27
|
iseri |
|- .~ Er ( S X. S ) |