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Theorem natrcl 14034
Description: Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
natrcl.1  |-  N  =  ( C Nat  D )
Assertion
Ref Expression
natrcl  |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )

Proof of Theorem natrcl
Dummy variables  x  f  y  a  g  h  r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natrcl.1 . . 3  |-  N  =  ( C Nat  D )
2 eqid 2366 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2366 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2366 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
5 eqid 2366 . . 3  |-  (comp `  D )  =  (comp `  D )
61, 2, 3, 4, 5natfval 14030 . 2  |-  N  =  ( f  e.  ( C  Func  D ) ,  g  e.  ( C  Func  D )  |->  [_ ( 1st `  f )  /  r ]_ [_ ( 1st `  g )  / 
s ]_ { a  e.  X_ x  e.  ( Base `  C ) ( ( r `  x
) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. h  e.  ( x
(  Hom  `  C ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
76elmpt2cl 6188 1  |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   {crab 2632   [_csb 3167   <.cop 3732   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   X_cixp 6960   Basecbs 13356    Hom chom 13427  compcco 13428    Func cfunc 13938   Nat cnat 14025
This theorem is referenced by:  nat1st2nd  14035  natixp  14036  nati  14039  fucco  14046  fuccocl  14048  fuclid  14050  fucrid  14051  fucass  14052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-ixp 6961  df-func 13942  df-nat 14027
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