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Theorem catcval 15281
Description: Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catcval.c  |-  C  =  (CatCat `  U )
catcval.u  |-  ( ph  ->  U  e.  V )
catcval.b  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
catcval.h  |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
catcval.o  |-  ( ph  ->  .x.  =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) ) )
Assertion
Ref Expression
catcval  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Distinct variable groups:    x, v,
y, z, B    ph, v, x, y, z    v, U, x, y, z    f,
g, v, x, y, z
Allowed substitution hints:    ph( f, g)    B( f, g)    C( x, y, z, v, f, g)    .x. ( x, y, z, v, f, g)    U( f, g)    H( x, y, z, v, f, g)    V( x, y, z, v, f, g)

Proof of Theorem catcval
Dummy variables  u  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcval.c . 2  |-  C  =  (CatCat `  U )
2 df-catc 15280 . . . 4  |- CatCat  =  ( u  e.  _V  |->  [_ ( u  i^i  Cat )  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  Func  y ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } )
32a1i 11 . . 3  |-  ( ph  -> CatCat 
=  ( u  e. 
_V  |->  [_ ( u  i^i 
Cat )  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  Func  y ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. } ) )
4 vex 3116 . . . . . 6  |-  u  e. 
_V
54inex1 4588 . . . . 5  |-  ( u  i^i  Cat )  e. 
_V
65a1i 11 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  e.  _V )
7 simpr 461 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
87ineq1d 3699 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  ( U  i^i  Cat ) )
9 catcval.b . . . . . 6  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
109adantr 465 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  B  =  ( U  i^i  Cat ) )
118, 10eqtr4d 2511 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  B )
12 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  b  =  B )
1312opeq2d 4220 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
14 eqidd 2468 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  Func  y )  =  ( x  Func  y ) )
1512, 12, 14mpt2eq123dv 6343 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( x  Func  y )
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
16 catcval.h . . . . . . . 8  |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x 
Func  y ) ) )
1716ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )
1815, 17eqtr4d 2511 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( x  Func  y )
)  =  H )
1918opeq2d 4220 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  Func  y
) ) >.  =  <. ( Hom  `  ndx ) ,  H >. )
2012, 12xpeq12d 5024 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
21 eqidd 2468 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) )  =  ( g  e.  ( ( 2nd `  v
)  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
2220, 12, 21mpt2eq123dv 6343 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )  =  ( v  e.  ( B  X.  B
) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v
)  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) ) )
23 catcval.o . . . . . . . 8  |-  ( ph  ->  .x.  =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z
) ,  f  e.  (  Func  `  v ) 
|->  ( g  o.func  f )
) ) )
2423ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  .x.  =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v
)  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) ) )
2522, 24eqtr4d 2511 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )  =  .x.  )
2625opeq2d 4220 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. (comp ` 
ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>.  =  <. (comp `  ndx ) ,  .x.  >. )
2713, 19, 26tpeq123d 4121 . . . 4  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  { <. (
Base `  ndx ) ,  b >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  Func  y
) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. }  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
286, 11, 27csbied2 3463 . . 3  |-  ( (
ph  /\  u  =  U )  ->  [_ (
u  i^i  Cat )  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  Func  y ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v ) 
Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) )
>. }  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
29 catcval.u . . . 4  |-  ( ph  ->  U  e.  V )
30 elex 3122 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
3129, 30syl 16 . . 3  |-  ( ph  ->  U  e.  _V )
32 tpex 6583 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  e.  _V
3332a1i 11 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. }  e.  _V )
343, 28, 31, 33fvmptd 5955 . 2  |-  ( ph  ->  (CatCat `  U )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
351, 34syl5eq 2520 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435    i^i cin 3475   {ctp 4031   <.cop 4033    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   2ndc2nd 6783   ndxcnx 14487   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919    Func cfunc 15081    o.func ccofu 15083  CatCatccatc 15279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-oprab 6288  df-mpt2 6289  df-catc 15280
This theorem is referenced by:  catcbas  15282  catchomfval  15283  catccofval  15285
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