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Theorem setcval 13925
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 13924 . . . 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 3819 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  u >.  =  <. (
Base `  ndx ) ,  U >. )
6 eqidd 2297 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
y  ^m  x )  =  ( y  ^m  x ) )
74, 4, 6mpt2eq123dv 5926 . . . . . 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 2331 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) )  =  H )
1110opeq2d 3819 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. (  Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x
) ) >.  =  <. (  Hom  `  ndx ) ,  H >. )
124, 4xpeq12d 4730 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
u  X.  u )  =  ( U  X.  U ) )
13 eqidd 2297 . . . . . . 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 5926 . . . . . 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 2331 . . . . 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 3819 . . . 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 3734 . . 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 2809 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
2220, 21syl 15 . . 3  |-  ( ph  ->  U  e.  _V )
23 tpex 4535 . . . 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 5622 . 2  |-  ( ph  ->  ( SetCat `  U )  =  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
261, 25syl5eq 2340 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 1632    e. wcel 1696   _Vcvv 2801   {ctp 3655   <.cop 3656    e. cmpt 4093    X. cxp 4703    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788   ndxcnx 13161   Basecbs 13164    Hom chom 13235  compcco 13236   SetCatcsetc 13923
This theorem is referenced by:  setcbas  13926  setchomfval  13927  setccofval  13930
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-setc 13924
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