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Theorem setcval 14161
Description: Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcval.c  |-  C  =  ( SetCat `  U )
setcval.u  |-  ( ph  ->  U  e.  V )
setcval.h  |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
setcval.o  |-  ( ph  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v
)  ^m  ( 1st `  v ) )  |->  ( g  o.  f ) ) ) )
Assertion
Ref Expression
setcval  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Distinct variable groups:    f, g,
v, x, y, z    ph, v, x, y, z   
v, U, x, y, z
Allowed substitution hints:    ph( 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 setcval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 setcval.c . 2  |-  C  =  ( SetCat `  U )
2 df-setc 14160 . . . 4  |-  SetCat  =  ( u  e.  _V  |->  {
<. ( Base `  ndx ) ,  u >. , 
<. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u
) ,  z  e.  u  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v
)  ^m  ( 1st `  v ) )  |->  ( g  o.  f ) ) ) >. } )
32a1i 11 . . 3  |-  ( ph  -> 
SetCat  =  ( u  e. 
_V  |->  { <. ( Base `  ndx ) ,  u >. ,  <. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x
) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) ) >. } ) )
4 simpr 448 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
54opeq2d 3935 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  u >.  =  <. (
Base `  ndx ) ,  U >. )
6 eqidd 2390 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
y  ^m  x )  =  ( y  ^m  x ) )
74, 4, 6mpt2eq123dv 6077 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) )  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
8 setcval.h . . . . . . 7  |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
107, 9eqtr4d 2424 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) )  =  H )
1110opeq2d 3935 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x
) ) >.  =  <. (  Hom  `  ndx ) ,  H >. )
124, 4xpeq12d 4845 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
u  X.  u )  =  ( U  X.  U ) )
13 eqidd 2390 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) )  =  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) )
1412, 4, 13mpt2eq123dv 6077 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  (
v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) )  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v
) )  |->  ( g  o.  f ) ) ) )
15 setcval.o . . . . . . 7  |-  ( ph  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v
)  ^m  ( 1st `  v ) )  |->  ( g  o.  f ) ) ) )
1615adantr 452 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v
) )  |->  ( g  o.  f ) ) ) )
1714, 16eqtr4d 2424 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) )  = 
.x.  )
1817opeq2d 3935 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. (comp ` 
ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) ) >.  =  <. (comp `  ndx ) ,  .x.  >. )
195, 11, 18tpeq123d 3843 . . 3  |-  ( (
ph  /\  u  =  U )  ->  { <. (
Base `  ndx ) ,  u >. ,  <. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x
) ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) ) >. }  =  { <. ( Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
20 setcval.u . . . 4  |-  ( ph  ->  U  e.  V )
21 elex 2909 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
2220, 21syl 16 . . 3  |-  ( ph  ->  U  e.  _V )
23 tpex 4650 . . . 4  |-  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  e.  _V
2423a1i 11 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. }  e.  _V )
253, 19, 22, 24fvmptd 5751 . 2  |-  ( ph  ->  ( SetCat `  U )  =  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
261, 25syl5eq 2433 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901   {ctp 3761   <.cop 3762    e. cmpt 4209    X. cxp 4818    o. ccom 4824   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   1stc1st 6288   2ndc2nd 6289    ^m cmap 6956   ndxcnx 13395   Basecbs 13398    Hom chom 13469  compcco 13470   SetCatcsetc 14159
This theorem is referenced by:  setcbas  14162  setchomfval  14163  setccofval  14166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-setc 14160
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