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Theorem natffn 14151
Description: The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypothesis
Ref Expression
natrcl.1  |-  N  =  ( C Nat  D )
Assertion
Ref Expression
natffn  |-  N  Fn  ( ( C  Func  D )  X.  ( C 
Func  D ) )

Proof of Theorem natffn
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 2438 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2438 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2438 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
5 eqid 2438 . . 3  |-  (comp `  D )  =  (comp `  D )
61, 2, 3, 4, 5natfval 14148 . 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 ) ) } )
7 fvex 5745 . . 3  |-  ( 1st `  f )  e.  _V
8 fvex 5745 . . . 4  |-  ( 1st `  g )  e.  _V
9 ovex 6109 . . . . . . 7  |-  ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  e. 
_V
109rgenw 2775 . . . . . 6  |-  A. x  e.  ( Base `  C
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  e.  _V
11 ixpexg 7089 . . . . . 6  |-  ( A. x  e.  ( Base `  C ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  e. 
_V  ->  X_ x  e.  (
Base `  C )
( ( r `  x ) (  Hom  `  D ) ( s `
 x ) )  e.  _V )
1210, 11ax-mp 5 . . . . 5  |-  X_ x  e.  ( Base `  C
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  e.  _V
1312rabex 4357 . . . 4  |-  { 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 ) ) }  e.  _V
148, 13csbex 3264 . . 3  |-  [_ ( 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 ) ) }  e.  _V
157, 14csbex 3264 . 2  |-  [_ ( 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 ) ) }  e.  _V
166, 15fnmpt2i 6423 1  |-  N  Fn  ( ( C  Func  D )  X.  ( C 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958   [_csb 3253   <.cop 3819    X. cxp 4879    Fn wfn 5452   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   X_cixp 7066   Basecbs 13474    Hom chom 13545  compcco 13546    Func cfunc 14056   Nat cnat 14143
This theorem is referenced by:  fuchom  14163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-ixp 7067  df-func 14060  df-nat 14145
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