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Theorem natffn 13922
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 2358 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2358 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2358 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
5 eqid 2358 . . 3  |-  (comp `  D )  =  (comp `  D )
61, 2, 3, 4, 5natfval 13919 . 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 5622 . . 3  |-  ( 1st `  f )  e.  _V
8 fvex 5622 . . . 4  |-  ( 1st `  g )  e.  _V
9 ovex 5970 . . . . . . 7  |-  ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  e. 
_V
109rgenw 2686 . . . . . 6  |-  A. x  e.  ( Base `  C
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  e.  _V
11 ixpexg 6928 . . . . . 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 8 . . . . 5  |-  X_ x  e.  ( Base `  C
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  e.  _V
1312rabex 4246 . . . 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 3168 . . 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 3168 . 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 6280 1  |-  N  Fn  ( ( C  Func  D )  X.  ( C 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864   [_csb 3157   <.cop 3719    X. cxp 4769    Fn wfn 5332   ` cfv 5337  (class class class)co 5945   1stc1st 6207   2ndc2nd 6208   X_cixp 6905   Basecbs 13245    Hom chom 13316  compcco 13317    Func cfunc 13827   Nat cnat 13914
This theorem is referenced by:  fuchom  13934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-ixp 6906  df-func 13831  df-nat 13916
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