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Theorem natffn 14109
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 2412 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2412 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2412 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
5 eqid 2412 . . 3  |-  (comp `  D )  =  (comp `  D )
61, 2, 3, 4, 5natfval 14106 . 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 5709 . . 3  |-  ( 1st `  f )  e.  _V
8 fvex 5709 . . . 4  |-  ( 1st `  g )  e.  _V
9 ovex 6073 . . . . . . 7  |-  ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  e. 
_V
109rgenw 2741 . . . . . 6  |-  A. x  e.  ( Base `  C
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  e.  _V
11 ixpexg 7053 . . . . . 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 4322 . . . 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 3230 . . 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 3230 . 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 6387 1  |-  N  Fn  ( ( C  Func  D )  X.  ( C 
Func  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924   [_csb 3219   <.cop 3785    X. cxp 4843    Fn wfn 5416   ` cfv 5421  (class class class)co 6048   1stc1st 6314   2ndc2nd 6315   X_cixp 7030   Basecbs 13432    Hom chom 13503  compcco 13504    Func cfunc 14014   Nat cnat 14101
This theorem is referenced by:  fuchom  14121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-ixp 7031  df-func 14018  df-nat 14103
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