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Theorem cofuval2 14089
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval2.b  |-  B  =  ( Base `  C
)
cofuval2.f  |-  ( ph  ->  F ( C  Func  D ) G )
cofuval2.x  |-  ( ph  ->  H ( D  Func  E ) K )
Assertion
Ref Expression
cofuval2  |-  ( ph  ->  ( <. H ,  K >.  o.func 
<. F ,  G >. )  =  <. ( H  o.  F ) ,  ( x  e.  B , 
y  e.  B  |->  ( ( ( F `  x ) K ( F `  y ) )  o.  ( x G y ) ) ) >. )
Distinct variable groups:    x, y, B    x, F, y    x, G, y    x, H, y    ph, x, y    x, K, y
Allowed substitution hints:    C( x, y)    D( x, y)    E( x, y)

Proof of Theorem cofuval2
StepHypRef Expression
1 cofuval2.b . . 3  |-  B  =  ( Base `  C
)
2 cofuval2.f . . . 4  |-  ( ph  ->  F ( C  Func  D ) G )
3 df-br 4216 . . . 4  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
42, 3sylib 190 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
5 cofuval2.x . . . 4  |-  ( ph  ->  H ( D  Func  E ) K )
6 df-br 4216 . . . 4  |-  ( H ( D  Func  E
) K  <->  <. H ,  K >.  e.  ( D 
Func  E ) )
75, 6sylib 190 . . 3  |-  ( ph  -> 
<. H ,  K >.  e.  ( D  Func  E
) )
81, 4, 7cofuval 14084 . 2  |-  ( ph  ->  ( <. H ,  K >.  o.func 
<. F ,  G >. )  =  <. ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. ) ) ,  ( x  e.  B , 
y  e.  B  |->  ( ( ( ( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. )
( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. ) y ) ) ) >. )
9 relfunc 14064 . . . . . 6  |-  Rel  ( D  Func  E )
10 brrelex12 4918 . . . . . 6  |-  ( ( Rel  ( D  Func  E )  /\  H ( D  Func  E ) K )  ->  ( H  e.  _V  /\  K  e.  _V ) )
119, 5, 10sylancr 646 . . . . 5  |-  ( ph  ->  ( H  e.  _V  /\  K  e.  _V )
)
12 op1stg 6362 . . . . 5  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 1st `  <. H ,  K >. )  =  H )
1311, 12syl 16 . . . 4  |-  ( ph  ->  ( 1st `  <. H ,  K >. )  =  H )
14 relfunc 14064 . . . . . 6  |-  Rel  ( C  Func  D )
15 brrelex12 4918 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F ( C  Func  D ) G )  ->  ( F  e.  _V  /\  G  e.  _V ) )
1614, 2, 15sylancr 646 . . . . 5  |-  ( ph  ->  ( F  e.  _V  /\  G  e.  _V )
)
17 op1stg 6362 . . . . 5  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. F ,  G >. )  =  F )
1816, 17syl 16 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
1913, 18coeq12d 5040 . . 3  |-  ( ph  ->  ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. )
)  =  ( H  o.  F ) )
20 op2ndg 6363 . . . . . . . 8  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 2nd `  <. H ,  K >. )  =  K )
2111, 20syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. H ,  K >. )  =  K )
22213ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. H ,  K >. )  =  K )
23183ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 1st ` 
<. F ,  G >. )  =  F )
2423fveq1d 5733 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  x )  =  ( F `  x ) )
2523fveq1d 5733 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  y )  =  ( F `  y ) )
2622, 24, 25oveq123d 6105 . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( (
( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. ) ( ( 1st `  <. F ,  G >. ) `  y ) )  =  ( ( F `  x ) K ( F `  y ) ) )
27 op2ndg 6363 . . . . . . . 8  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. F ,  G >. )  =  G )
2816, 27syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
29283ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. F ,  G >. )  =  G )
3029oveqd 6101 . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x
( 2nd `  <. F ,  G >. )
y )  =  ( x G y ) )
3126, 30coeq12d 5040 . . . 4  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( (
( ( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. ) ( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. )
y ) )  =  ( ( ( F `
 x ) K ( F `  y
) )  o.  (
x G y ) ) )
3231mpt2eq3dva 6141 . . 3  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  <. F ,  G >. ) `  x
) ( 2nd `  <. H ,  K >. )
( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. ) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( ( ( F `  x
) K ( F `
 y ) )  o.  ( x G y ) ) ) )
3319, 32opeq12d 3994 . 2  |-  ( ph  -> 
<. ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. )
) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  <. F ,  G >. ) `  x ) ( 2nd `  <. H ,  K >. ) ( ( 1st `  <. F ,  G >. ) `  y ) )  o.  ( x ( 2nd `  <. F ,  G >. )
y ) ) )
>.  =  <. ( H  o.  F ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( F `
 x ) K ( F `  y
) )  o.  (
x G y ) ) ) >. )
348, 33eqtrd 2470 1  |-  ( ph  ->  ( <. H ,  K >.  o.func 
<. F ,  G >. )  =  <. ( H  o.  F ) ,  ( x  e.  B , 
y  e.  B  |->  ( ( ( F `  x ) K ( F `  y ) )  o.  ( x G y ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4215    o. ccom 4885   Rel wrel 4886   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Func cfunc 14056    o.func ccofu 14058
This theorem is referenced by:  catcisolem  14266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023  df-ixp 7067  df-func 14060  df-cofu 14062
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