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Theorem catcval 16046
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 16045 . . . 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 3060 . . . . . 6  |-  u  e. 
_V
54inex1 4560 . . . . 5  |-  ( u  i^i  Cat )  e. 
_V
65a1i 11 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  e.  _V )
7 simpr 467 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
87ineq1d 3645 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  ( U  i^i  Cat ) )
9 catcval.b . . . . . 6  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
109adantr 471 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  B  =  ( U  i^i  Cat ) )
118, 10eqtr4d 2499 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  B )
12 simpr 467 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  b  =  B )
1312opeq2d 4187 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
14 eqidd 2463 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  Func  y )  =  ( x  Func  y ) )
1512, 12, 14mpt2eq123dv 6385 . . . . . . 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 737 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )
1815, 17eqtr4d 2499 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( x  Func  y )
)  =  H )
1918opeq2d 4187 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  Func  y
) ) >.  =  <. ( Hom  `  ndx ) ,  H >. )
2012sqxpeqd 4882 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
21 eqidd 2463 . . . . . . . 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 6385 . . . . . . 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 737 . . . . . . 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 2499 . . . . . 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 4187 . . . . 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 4079 . . . 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 3403 . . 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 3066 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
3129, 30syl 17 . . 3  |-  ( ph  ->  U  e.  _V )
32 tpex 6622 . . . 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 5982 . 2  |-  ( ph  ->  (CatCat `  U )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
351, 34syl5eq 2508 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057   [_csb 3375    i^i cin 3415   {ctp 3984   <.cop 3986    |-> cmpt 4477    X. cxp 4854   ` cfv 5605  (class class class)co 6320    |-> cmpt2 6322   2ndc2nd 6824   ndxcnx 15173   Basecbs 15176   Hom chom 15256  compcco 15257   Catccat 15625    Func cfunc 15814    o.func ccofu 15816  CatCatccatc 16044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-oprab 6324  df-mpt2 6325  df-catc 16045
This theorem is referenced by:  catcbas  16047  catchomfval  16048  catccofval  16050
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