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Theorem fnxpc 14265
Description: The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
Assertion
Ref Expression
fnxpc  |-  X.c  Fn  ( _V  X.  _V )

Proof of Theorem fnxpc
Dummy variables  f 
b  g  h  r  s  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xpc 14261 . 2  |-  X.c  =  ( r  e.  _V , 
s  e.  _V  |->  [_ ( ( Base `  r
)  X.  ( Base `  s ) )  / 
b ]_ [_ ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) (  Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. (  Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
2 fvex 5734 . . . 4  |-  ( Base `  r )  e.  _V
3 fvex 5734 . . . 4  |-  ( Base `  s )  e.  _V
42, 3xpex 4982 . . 3  |-  ( (
Base `  r )  X.  ( Base `  s
) )  e.  _V
5 vex 2951 . . . . 5  |-  b  e. 
_V
65, 5mpt2ex 6417 . . . 4  |-  ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) (  Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  s
) ( 2nd `  v
) ) ) )  e.  _V
7 tpex 4700 . . . 4  |-  { <. (
Base `  ndx ) ,  b >. ,  <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  e.  _V
86, 7csbex 3254 . . 3  |-  [_ (
u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u
) (  Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. (  Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  e.  _V
94, 8csbex 3254 . 2  |-  [_ (
( Base `  r )  X.  ( Base `  s
) )  /  b ]_ [_ ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u ) (  Hom  `  r
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  s
) ( 2nd `  v
) ) ) )  /  h ]_ { <. ( Base `  ndx ) ,  b >. , 
<. (  Hom  `  ndx ) ,  h >. , 
<. (comp `  ndx ) ,  ( x  e.  ( b  X.  b ) ,  y  e.  b 
|->  ( g  e.  ( ( 2nd `  x
) h y ) ,  f  e.  ( h `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  r )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  s )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  e.  _V
101, 9fnmpt2i 6412 1  |-  X.c  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2948   [_csb 3243   {ctp 3808   <.cop 3809    X. cxp 4868    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   ndxcnx 13458   Basecbs 13461    Hom chom 13532  compcco 13533    X.c cxpc 14257
This theorem is referenced by:  xpcbas  14267  xpchomfval  14268  xpccofval  14271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-xpc 14261
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