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Theorem cofuval2 13761
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 4024 . . . 4  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
42, 3sylib 188 . . 3  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
5 cofuval2.x . . . 4  |-  ( ph  ->  H ( D  Func  E ) K )
6 df-br 4024 . . . 4  |-  ( H ( D  Func  E
) K  <->  <. H ,  K >.  e.  ( D 
Func  E ) )
75, 6sylib 188 . . 3  |-  ( ph  -> 
<. H ,  K >.  e.  ( D  Func  E
) )
81, 4, 7cofuval 13756 . 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 13736 . . . . . 6  |-  Rel  ( D  Func  E )
10 brrelex12 4726 . . . . . 6  |-  ( ( Rel  ( D  Func  E )  /\  H ( D  Func  E ) K )  ->  ( H  e.  _V  /\  K  e.  _V ) )
119, 5, 10sylancr 644 . . . . 5  |-  ( ph  ->  ( H  e.  _V  /\  K  e.  _V )
)
12 op1stg 6132 . . . . 5  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 1st `  <. H ,  K >. )  =  H )
1311, 12syl 15 . . . 4  |-  ( ph  ->  ( 1st `  <. H ,  K >. )  =  H )
14 relfunc 13736 . . . . . 6  |-  Rel  ( C  Func  D )
15 brrelex12 4726 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F ( C  Func  D ) G )  ->  ( F  e.  _V  /\  G  e.  _V ) )
1614, 2, 15sylancr 644 . . . . 5  |-  ( ph  ->  ( F  e.  _V  /\  G  e.  _V )
)
17 op1stg 6132 . . . . 5  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 1st `  <. F ,  G >. )  =  F )
1816, 17syl 15 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
1913, 18coeq12d 4848 . . 3  |-  ( ph  ->  ( ( 1st `  <. H ,  K >. )  o.  ( 1st `  <. F ,  G >. )
)  =  ( H  o.  F ) )
20 op2ndg 6133 . . . . . . . 8  |-  ( ( H  e.  _V  /\  K  e.  _V )  ->  ( 2nd `  <. H ,  K >. )  =  K )
2111, 20syl 15 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. H ,  K >. )  =  K )
22213ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. H ,  K >. )  =  K )
23183ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 1st ` 
<. F ,  G >. )  =  F )
2423fveq1d 5527 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  x )  =  ( F `  x ) )
2523fveq1d 5527 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( ( 1st `  <. F ,  G >. ) `  y )  =  ( F `  y ) )
2622, 24, 25oveq123d 5879 . . . . 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 6133 . . . . . . . 8  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( 2nd `  <. F ,  G >. )  =  G )
2816, 27syl 15 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
29283ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( 2nd ` 
<. F ,  G >. )  =  G )
3029oveqd 5875 . . . . 5  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x
( 2nd `  <. F ,  G >. )
y )  =  ( x G y ) )
3126, 30coeq12d 4848 . . . 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 5912 . . 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 3804 . 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 2315 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    o. ccom 4693   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Func cfunc 13728    o.func ccofu 13730
This theorem is referenced by:  catcisolem  13938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-cofu 13734
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