MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setcval Unicode version

Theorem setcval 13909
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 13908 . . . 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 10 . . 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 447 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
54opeq2d 3803 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  u >.  =  <. (
Base `  ndx ) ,  U >. )
6 eqidd 2284 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
y  ^m  x )  =  ( y  ^m  x ) )
74, 4, 6mpt2eq123dv 5910 . . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
107, 9eqtr4d 2318 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) )  =  H )
1110opeq2d 3803 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x
) ) >.  =  <. (  Hom  `  ndx ) ,  H >. )
124, 4xpeq12d 4714 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
u  X.  u )  =  ( U  X.  U ) )
13 eqidd 2284 . . . . . . 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 5910 . . . . . 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 451 . . . . . 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 2318 . . . . 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 3803 . . . 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 3721 . . 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 2796 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
2220, 21syl 15 . . 3  |-  ( ph  ->  U  e.  _V )
23 tpex 4519 . . . 4  |-  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  e.  _V
2423a1i 10 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. }  e.  _V )
253, 19, 22, 24fvmptd 5606 . 2  |-  ( ph  ->  ( SetCat `  U )  =  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
261, 25syl5eq 2327 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {ctp 3642   <.cop 3643    e. cmpt 4077    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   ndxcnx 13145   Basecbs 13148    Hom chom 13219  compcco 13220   SetCatcsetc 13907
This theorem is referenced by:  setcbas  13910  setchomfval  13911  setccofval  13914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-setc 13908
  Copyright terms: Public domain W3C validator