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Theorem natffn 13823
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 2283 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2283 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
5 eqid 2283 . . 3  |-  (comp `  D )  =  (comp `  D )
61, 2, 3, 4, 5natfval 13820 . 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 5539 . . 3  |-  ( 1st `  f )  e.  _V
8 fvex 5539 . . . 4  |-  ( 1st `  g )  e.  _V
9 ovex 5883 . . . . . . 7  |-  ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  e. 
_V
109rgenw 2610 . . . . . 6  |-  A. x  e.  ( Base `  C
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  e.  _V
11 ixpexg 6840 . . . . . 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 4165 . . . 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 3092 . . 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 3092 . 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 6193 1  |-  N  Fn  ( ( C  Func  D )  X.  ( C 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   [_csb 3081   <.cop 3643    X. cxp 4687    Fn wfn 5250   ` 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:  fuchom  13835
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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