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Theorem catcval 13944
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 13943 . . . 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 10 . . 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 2804 . . . . . 6  |-  u  e. 
_V
54inex1 4171 . . . . 5  |-  ( u  i^i  Cat )  e. 
_V
65a1i 10 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  e.  _V )
7 simpr 447 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
87ineq1d 3382 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  ( U  i^i  Cat ) )
9 catcval.b . . . . . 6  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
109adantr 451 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  B  =  ( U  i^i  Cat ) )
118, 10eqtr4d 2331 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  B )
12 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  b  =  B )
1312opeq2d 3819 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
14 eqidd 2297 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  Func  y )  =  ( x  Func  y ) )
1512, 12, 14mpt2eq123dv 5926 . . . . . . 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 706 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )
1815, 17eqtr4d 2331 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( x  Func  y )
)  =  H )
1918opeq2d 3819 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. (  Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  Func  y
) ) >.  =  <. (  Hom  `  ndx ) ,  H >. )
2012, 12xpeq12d 4730 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
21 eqidd 2297 . . . . . . . 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 5926 . . . . . . 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 706 . . . . . . 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 2331 . . . . . 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 3819 . . . . 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 3734 . . . 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 3137 . . 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 2809 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
3129, 30syl 15 . . 3  |-  ( ph  ->  U  e.  _V )
32 tpex 4535 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  e.  _V
3332a1i 10 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. }  e.  _V )
343, 28, 31, 33fvmptd 5622 . 2  |-  ( ph  ->  (CatCat `  U )  =  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
351, 34syl5eq 2340 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094    i^i cin 3164   {ctp 3655   <.cop 3656    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   ndxcnx 13161   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582    Func cfunc 13744    o.func ccofu 13746  CatCatccatc 13942
This theorem is referenced by:  catcbas  13945  catchomfval  13946  catccofval  13948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-catc 13943
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