Step |
Hyp |
Ref |
Expression |
1 |
|
catcbas.c |
|- C = ( CatCat ` U ) |
2 |
|
catcbas.b |
|- B = ( Base ` C ) |
3 |
|
catcbas.u |
|- ( ph -> U e. V ) |
4 |
|
catcco.o |
|- .x. = ( comp ` C ) |
5 |
|
catcco.x |
|- ( ph -> X e. B ) |
6 |
|
catcco.y |
|- ( ph -> Y e. B ) |
7 |
|
catcco.z |
|- ( ph -> Z e. B ) |
8 |
|
catcco.f |
|- ( ph -> F e. ( X Func Y ) ) |
9 |
|
catcco.g |
|- ( ph -> G e. ( Y Func Z ) ) |
10 |
1 2 3 4
|
catccofval |
|- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) ) |
11 |
|
simprl |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> v = <. X , Y >. ) |
12 |
11
|
fveq2d |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` v ) = ( 2nd ` <. X , Y >. ) ) |
13 |
|
op2ndg |
|- ( ( X e. B /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y ) |
14 |
5 6 13
|
syl2anc |
|- ( ph -> ( 2nd ` <. X , Y >. ) = Y ) |
15 |
14
|
adantr |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` <. X , Y >. ) = Y ) |
16 |
12 15
|
eqtrd |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` v ) = Y ) |
17 |
|
simprr |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> z = Z ) |
18 |
16 17
|
oveq12d |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( ( 2nd ` v ) Func z ) = ( Y Func Z ) ) |
19 |
11
|
fveq2d |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Func ` v ) = ( Func ` <. X , Y >. ) ) |
20 |
|
df-ov |
|- ( X Func Y ) = ( Func ` <. X , Y >. ) |
21 |
19 20
|
eqtr4di |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( Func ` v ) = ( X Func Y ) ) |
22 |
|
eqidd |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( g o.func f ) = ( g o.func f ) ) |
23 |
18 21 22
|
mpoeq123dv |
|- ( ( ph /\ ( v = <. X , Y >. /\ z = Z ) ) -> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) = ( g e. ( Y Func Z ) , f e. ( X Func Y ) |-> ( g o.func f ) ) ) |
24 |
5 6
|
opelxpd |
|- ( ph -> <. X , Y >. e. ( B X. B ) ) |
25 |
|
ovex |
|- ( Y Func Z ) e. _V |
26 |
|
ovex |
|- ( X Func Y ) e. _V |
27 |
25 26
|
mpoex |
|- ( g e. ( Y Func Z ) , f e. ( X Func Y ) |-> ( g o.func f ) ) e. _V |
28 |
27
|
a1i |
|- ( ph -> ( g e. ( Y Func Z ) , f e. ( X Func Y ) |-> ( g o.func f ) ) e. _V ) |
29 |
10 23 24 7 28
|
ovmpod |
|- ( ph -> ( <. X , Y >. .x. Z ) = ( g e. ( Y Func Z ) , f e. ( X Func Y ) |-> ( g o.func f ) ) ) |
30 |
|
oveq12 |
|- ( ( g = G /\ f = F ) -> ( g o.func f ) = ( G o.func F ) ) |
31 |
30
|
adantl |
|- ( ( ph /\ ( g = G /\ f = F ) ) -> ( g o.func f ) = ( G o.func F ) ) |
32 |
|
ovexd |
|- ( ph -> ( G o.func F ) e. _V ) |
33 |
29 31 9 8 32
|
ovmpod |
|- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o.func F ) ) |