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Theorem isnat2 13822
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 13736 . . . . 5  |-  Rel  ( C  Func  D )
2 isnat2.f . . . . 5  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 1st2nd 6166 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
41, 2, 3sylancr 644 . . . 4  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
5 isnat2.g . . . . 5  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
6 1st2nd 6166 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
71, 5, 6sylancr 644 . . . 4  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
84, 7oveq12d 5876 . . 3  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
98eleq2d 2350 . 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 6169 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
161, 2, 15sylancr 644 . . 3  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
17 1st2ndbr 6169 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
181, 5, 17sylancr 644 . . 3  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
1910, 11, 12, 13, 14, 16, 18isnat 13821 . 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 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   X_cixp 6817   Basecbs 13148    Hom chom 13219  compcco 13220    Func cfunc 13728   Nat cnat 13815
This theorem is referenced by:  fuccocl  13838  fucidcl  13839  invfuc  13848  curf2cl  14005  yonedalem4c  14051  yonedalem3  14054
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-ixp 6818  df-func 13732  df-nat 13817
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