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Theorem cofuval2 14047
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 4181 . . . 4  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
42, 3sylib 189 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
5 cofuval2.x . . . 4  |-  ( ph  ->  H ( D  Func  E ) K )
6 df-br 4181 . . . 4  |-  ( H ( D  Func  E
) K  <->  <. H ,  K >.  e.  ( D 
Func  E ) )
75, 6sylib 189 . . 3  |-  ( ph  -> 
<. H ,  K >.  e.  ( D  Func  E
) )
81, 4, 7cofuval 14042 . 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 14022 . . . . . 6  |-  Rel  ( D  Func  E )
10 brrelex12 4882 . . . . . 6  |-  ( ( Rel  ( D  Func  E )  /\  H ( D  Func  E ) K )  ->  ( H  e.  _V  /\  K  e.  _V ) )
119, 5, 10sylancr 645 . . . . 5  |-  ( ph  ->  ( H  e.  _V  /\  K  e.  _V )
)
12 op1stg 6326 . . . . 5  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 1st `  <. H ,  K >. )  =  H )
1311, 12syl 16 . . . 4  |-  ( ph  ->  ( 1st `  <. H ,  K >. )  =  H )
14 relfunc 14022 . . . . . 6  |-  Rel  ( C  Func  D )
15 brrelex12 4882 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F ( C  Func  D ) G )  ->  ( F  e.  _V  /\  G  e.  _V ) )
1614, 2, 15sylancr 645 . . . . 5  |-  ( ph  ->  ( F  e.  _V  /\  G  e.  _V )
)
17 op1stg 6326 . . . . 5  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. F ,  G >. )  =  F )
1816, 17syl 16 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
1913, 18coeq12d 5004 . . 3  |-  ( ph  ->  ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. )
)  =  ( H  o.  F ) )
20 op2ndg 6327 . . . . . . . 8  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 2nd `  <. H ,  K >. )  =  K )
2111, 20syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. H ,  K >. )  =  K )
22213ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. H ,  K >. )  =  K )
23183ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 1st ` 
<. F ,  G >. )  =  F )
2423fveq1d 5697 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  x )  =  ( F `  x ) )
2523fveq1d 5697 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  y )  =  ( F `  y ) )
2622, 24, 25oveq123d 6069 . . . . 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 6327 . . . . . . . 8  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. F ,  G >. )  =  G )
2816, 27syl 16 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
29283ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. F ,  G >. )  =  G )
3029oveqd 6065 . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x
( 2nd `  <. F ,  G >. )
y )  =  ( x G y ) )
3126, 30coeq12d 5004 . . . 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 6105 . . 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 3960 . 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 2444 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785   class class class wbr 4180    o. ccom 4849   Rel wrel 4850   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1stc1st 6314   2ndc2nd 6315   Basecbs 13432    Func cfunc 14014    o.func ccofu 14016
This theorem is referenced by:  catcisolem  14224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-map 6987  df-ixp 7031  df-func 14018  df-cofu 14020
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