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Theorem isnat2 14145
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 14059 . . . . 5  |-  Rel  ( C  Func  D )
2 isnat2.f . . . . 5  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 1st2nd 6393 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
41, 2, 3sylancr 645 . . . 4  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
5 isnat2.g . . . . 5  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
6 1st2nd 6393 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
71, 5, 6sylancr 645 . . . 4  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
84, 7oveq12d 6099 . . 3  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
98eleq2d 2503 . 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 6396 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
161, 2, 15sylancr 645 . . 3  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
17 1st2ndbr 6396 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
181, 5, 17sylancr 645 . . 3  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
1910, 11, 12, 13, 14, 16, 18isnat 14144 . 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 245 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817   class class class wbr 4212   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   X_cixp 7063   Basecbs 13469    Hom chom 13540  compcco 13541    Func cfunc 14051   Nat cnat 14138
This theorem is referenced by:  fuccocl  14161  fucidcl  14162  invfuc  14171  curf2cl  14328  yonedalem4c  14374  yonedalem3  14377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-ixp 7064  df-func 14055  df-nat 14140
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