Step |
Hyp |
Ref |
Expression |
1 |
|
fvmpopr2d.1 |
|- ( ph -> F = ( a e. A , b e. B |-> C ) ) |
2 |
|
fvmpopr2d.2 |
|- ( ph -> P = <. a , b >. ) |
3 |
|
fvmpopr2d.3 |
|- ( ( ph /\ a e. A /\ b e. B ) -> C e. V ) |
4 |
|
df-ov |
|- ( a ( a e. A , b e. B |-> C ) b ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. ) |
5 |
1
|
3ad2ant1 |
|- ( ( ph /\ a e. A /\ b e. B ) -> F = ( a e. A , b e. B |-> C ) ) |
6 |
2
|
3ad2ant1 |
|- ( ( ph /\ a e. A /\ b e. B ) -> P = <. a , b >. ) |
7 |
5 6
|
fveq12d |
|- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. ) ) |
8 |
4 7
|
eqtr4id |
|- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = ( F ` P ) ) |
9 |
|
nfcv |
|- F/_ c C |
10 |
|
nfcv |
|- F/_ d C |
11 |
|
nfcv |
|- F/_ a d |
12 |
|
nfcsb1v |
|- F/_ a [_ c / a ]_ C |
13 |
11 12
|
nfcsbw |
|- F/_ a [_ d / b ]_ [_ c / a ]_ C |
14 |
|
nfcsb1v |
|- F/_ b [_ d / b ]_ [_ c / a ]_ C |
15 |
|
csbeq1a |
|- ( a = c -> C = [_ c / a ]_ C ) |
16 |
|
csbeq1a |
|- ( b = d -> [_ c / a ]_ C = [_ d / b ]_ [_ c / a ]_ C ) |
17 |
15 16
|
sylan9eq |
|- ( ( a = c /\ b = d ) -> C = [_ d / b ]_ [_ c / a ]_ C ) |
18 |
9 10 13 14 17
|
cbvmpo |
|- ( a e. A , b e. B |-> C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) |
19 |
18
|
oveqi |
|- ( a ( a e. A , b e. B |-> C ) b ) = ( a ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) b ) |
20 |
|
eqidd |
|- ( ( ph /\ a e. A /\ b e. B ) -> ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) ) |
21 |
|
equcom |
|- ( a = c <-> c = a ) |
22 |
|
equcom |
|- ( b = d <-> d = b ) |
23 |
21 22
|
anbi12i |
|- ( ( a = c /\ b = d ) <-> ( c = a /\ d = b ) ) |
24 |
23 17
|
sylbir |
|- ( ( c = a /\ d = b ) -> C = [_ d / b ]_ [_ c / a ]_ C ) |
25 |
24
|
eqcomd |
|- ( ( c = a /\ d = b ) -> [_ d / b ]_ [_ c / a ]_ C = C ) |
26 |
25
|
adantl |
|- ( ( ( ph /\ a e. A /\ b e. B ) /\ ( c = a /\ d = b ) ) -> [_ d / b ]_ [_ c / a ]_ C = C ) |
27 |
|
simp2 |
|- ( ( ph /\ a e. A /\ b e. B ) -> a e. A ) |
28 |
|
simp3 |
|- ( ( ph /\ a e. A /\ b e. B ) -> b e. B ) |
29 |
20 26 27 28 3
|
ovmpod |
|- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) b ) = C ) |
30 |
19 29
|
eqtrid |
|- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = C ) |
31 |
8 30
|
eqtr3d |
|- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C ) |