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Theorem isnat2 14857
Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natfval.1  |-  N  =  ( C Nat  D )
natfval.b  |-  B  =  ( Base `  C
)
natfval.h  |-  H  =  ( Hom  `  C
)
natfval.j  |-  J  =  ( Hom  `  D
)
natfval.o  |-  .x.  =  (comp `  D )
isnat2.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
isnat2.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
Assertion
Ref Expression
isnat2  |-  ( ph  ->  ( A  e.  ( F N G )  <-> 
( A  e.  X_ x  e.  B  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `  y
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >.  .x.  ( ( 1st `  G ) `  y ) ) ( ( x ( 2nd `  F ) y ) `
 h ) )  =  ( ( ( x ( 2nd `  G
) y ) `  h ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  G ) `  y ) ) ( A `  x ) ) ) ) )
Distinct variable groups:    x, h, y, A    x, B, y    C, h, x, y    h, F, x, y    h, G, x, y    h, H    ph, h, x, y    D, h, x, y
Allowed substitution hints:    B( h)    .x. ( x, y, h)    H( x, y)    J( x, y, h)    N( x, y, h)

Proof of Theorem isnat2
StepHypRef Expression
1 relfunc 14771 . . . . 5  |-  Rel  ( C  Func  D )
2 isnat2.f . . . . 5  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 1st2nd 6619 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
41, 2, 3sylancr 663 . . . 4  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
5 isnat2.g . . . . 5  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
6 1st2nd 6619 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
71, 5, 6sylancr 663 . . . 4  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
84, 7oveq12d 6108 . . 3  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
98eleq2d 2509 . 2  |-  ( ph  ->  ( A  e.  ( F N G )  <-> 
A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) ) )
10 natfval.1 . . 3  |-  N  =  ( C Nat  D )
11 natfval.b . . 3  |-  B  =  ( Base `  C
)
12 natfval.h . . 3  |-  H  =  ( Hom  `  C
)
13 natfval.j . . 3  |-  J  =  ( Hom  `  D
)
14 natfval.o . . 3  |-  .x.  =  (comp `  D )
15 1st2ndbr 6622 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
161, 2, 15sylancr 663 . . 3  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
17 1st2ndbr 6622 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
181, 5, 17sylancr 663 . . 3  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
1910, 11, 12, 13, 14, 16, 18isnat 14856 . 2  |-  ( ph  ->  ( A  e.  (
<. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. )  <->  ( A  e.  X_ x  e.  B  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) )  /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >.  .x.  ( ( 1st `  G ) `  y ) ) ( ( x ( 2nd `  F ) y ) `
 h ) )  =  ( ( ( x ( 2nd `  G
) y ) `  h ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  G ) `  y ) ) ( A `  x ) ) ) ) )
209, 19bitrd 253 1  |-  ( ph  ->  ( A  e.  ( F N G )  <-> 
( A  e.  X_ x  e.  B  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `  y
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >.  .x.  ( ( 1st `  G ) `  y ) ) ( ( x ( 2nd `  F ) y ) `
 h ) )  =  ( ( ( x ( 2nd `  G
) y ) `  h ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >.  .x.  ( ( 1st `  G ) `  y ) ) ( A `  x ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   <.cop 3882   class class class wbr 4291   Rel wrel 4844   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   X_cixp 7262   Basecbs 14173   Hom chom 14248  compcco 14249    Func cfunc 14763   Nat cnat 14850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-ixp 7263  df-func 14767  df-nat 14852
This theorem is referenced by:  fuccocl  14873  fucidcl  14874  invfuc  14883  curf2cl  15040  yonedalem4c  15086  yonedalem3  15089
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