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Theorem setcval 16027
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 16026 . . . 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 467 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
54opeq2d 4187 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  u >.  =  <. (
Base `  ndx ) ,  U >. )
6 eqidd 2463 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
y  ^m  x )  =  ( y  ^m  x ) )
74, 4, 6mpt2eq123dv 6385 . . . . . 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 471 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
107, 9eqtr4d 2499 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) )  =  H )
1110opeq2d 4187 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x
) ) >.  =  <. ( Hom  `  ndx ) ,  H >. )
124sqxpeqd 4882 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
u  X.  u )  =  ( U  X.  U ) )
13 eqidd 2463 . . . . . . 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 6385 . . . . . 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 471 . . . . . 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 2499 . . . . 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 4187 . . . 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 4079 . . 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 3066 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
2220, 21syl 17 . . 3  |-  ( ph  ->  U  e.  _V )
23 tpex 6622 . . . 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 5982 . 2  |-  ( ph  ->  ( SetCat `  U )  =  { <. ( Base `  ndx ) ,  U >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
261, 25syl5eq 2508 1  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  U >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057   {ctp 3984   <.cop 3986    |-> cmpt 4477    X. cxp 4854    o. ccom 4860   ` cfv 5605  (class class class)co 6320    |-> cmpt2 6322   1stc1st 6823   2ndc2nd 6824    ^m cmap 7503   ndxcnx 15173   Basecbs 15176   Hom chom 15256  compcco 15257   SetCatcsetc 16025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-oprab 6324  df-mpt2 6325  df-setc 16026
This theorem is referenced by:  setcbas  16028  setchomfval  16029  setccofval  16032
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