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Theorem catcval 14243
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 14242 . . . 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 2951 . . . . . 6  |-  u  e. 
_V
54inex1 4336 . . . . 5  |-  ( u  i^i  Cat )  e. 
_V
65a1i 11 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  e.  _V )
7 simpr 448 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
87ineq1d 3533 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  ( U  i^i  Cat ) )
9 catcval.b . . . . . 6  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
109adantr 452 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  B  =  ( U  i^i  Cat ) )
118, 10eqtr4d 2470 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  (
u  i^i  Cat )  =  B )
12 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  b  =  B )
1312opeq2d 3983 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
14 eqidd 2436 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  Func  y )  =  ( x  Func  y ) )
1512, 12, 14mpt2eq123dv 6128 . . . . . . 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 707 . . . . . . 7  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )
1815, 17eqtr4d 2470 . . . . . 6  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( x  Func  y )
)  =  H )
1918opeq2d 3983 . . . . 5  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  <. (  Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b 
|->  ( x  Func  y
) ) >.  =  <. (  Hom  `  ndx ) ,  H >. )
2012, 12xpeq12d 4895 . . . . . . . 8  |-  ( ( ( ph  /\  u  =  U )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
21 eqidd 2436 . . . . . . . 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 6128 . . . . . . 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 707 . . . . . . 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 2470 . . . . . 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 3983 . . . . 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 3890 . . . 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 3286 . . 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 2956 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
3129, 30syl 16 . . 3  |-  ( ph  ->  U  e.  _V )
32 tpex 4700 . . . 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 5802 . 2  |-  ( ph  ->  (CatCat `  U )  =  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
351, 34syl5eq 2479 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243    i^i cin 3311   {ctp 3808   <.cop 3809    e. cmpt 4258    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   2ndc2nd 6340   ndxcnx 13458   Basecbs 13461    Hom chom 13532  compcco 13533   Catccat 13881    Func cfunc 14043    o.func ccofu 14045  CatCatccatc 14241
This theorem is referenced by:  catcbas  14244  catchomfval  14245  catccofval  14247
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-sep 4322  ax-nul 4330  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-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-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  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-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-catc 14242
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