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Theorem estrcval 16020
Description: Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
Hypotheses
Ref Expression
estrcval.c  |-  C  =  (ExtStrCat `  U )
estrcval.u  |-  ( ph  ->  U  e.  V )
estrcval.h  |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( (
Base `  y )  ^m  ( Base `  x
) ) ) )
estrcval.o  |-  ( ph  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( ( Base `  z )  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) ) )
Assertion
Ref Expression
estrcval  |-  ( 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 estrcval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 estrcval.c . 2  |-  C  =  (ExtStrCat `  U )
2 df-estrc 16019 . . . 4  |- ExtStrCat  =  ( u  e.  _V  |->  {
<. ( Base `  ndx ) ,  u >. , 
<. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( (
Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  ( ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) )  |->  ( g  o.  f ) ) )
>. } )
32a1i 11 . . 3  |-  ( ph  -> ExtStrCat  =  ( u  e. 
_V  |->  { <. ( Base `  ndx ) ,  u >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( ( Base `  z )  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) ) >. } ) )
4 simpr 467 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  u  =  U )
54opeq2d 4143 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Base `  ndx ) ,  u >.  =  <. (
Base `  ndx ) ,  U >. )
6 eqidd 2453 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
( Base `  y )  ^m  ( Base `  x
) )  =  ( ( Base `  y
)  ^m  ( Base `  x ) ) )
74, 4, 6mpt2eq123dv 6341 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )  =  ( x  e.  U ,  y  e.  U  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) )
8 estrcval.h . . . . . . 7  |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( (
Base `  y )  ^m  ( Base `  x
) ) ) )
98adantr 471 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) )
107, 9eqtr4d 2489 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
x  e.  u ,  y  e.  u  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )  =  H )
1110opeq2d 4143 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )
>.  =  <. ( Hom  `  ndx ) ,  H >. )
124sqxpeqd 4838 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
u  X.  u )  =  ( U  X.  U ) )
13 eqidd 2453 . . . . . . 7  |-  ( (
ph  /\  u  =  U )  ->  (
g  e.  ( (
Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  ( ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) )  |->  ( g  o.  f ) )  =  ( g  e.  ( ( Base `  z
)  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) )
1412, 4, 13mpt2eq123dv 6341 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  (
v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( (
Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  ( ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) )  |->  ( g  o.  f ) ) )  =  ( v  e.  ( U  X.  U
) ,  z  e.  U  |->  ( g  e.  ( ( Base `  z
)  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) ) )
15 estrcval.o . . . . . . 7  |-  ( ph  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( ( Base `  z )  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) ) )
1615adantr 471 . . . . . 6  |-  ( (
ph  /\  u  =  U )  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( ( Base `  z
)  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) ) )
1714, 16eqtr4d 2489 . . . . 5  |-  ( (
ph  /\  u  =  U )  ->  (
v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( (
Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  ( ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) )  |->  ( g  o.  f ) ) )  =  .x.  )
1817opeq2d 4143 . . . 4  |-  ( (
ph  /\  u  =  U )  ->  <. (comp ` 
ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( (
Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  ( ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) )  |->  ( g  o.  f ) ) )
>.  =  <. (comp `  ndx ) ,  .x.  >. )
195, 11, 18tpeq123d 4035 . . 3  |-  ( (
ph  /\  u  =  U )  ->  { <. (
Base `  ndx ) ,  u >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( ( Base `  y
)  ^m  ( Base `  x ) ) )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( ( Base `  z )  ^m  ( Base `  ( 2nd `  v
) ) ) ,  f  e.  ( (
Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
|->  ( g  o.  f
) ) ) >. }  =  { <. ( Base `  ndx ) ,  U >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
20 estrcval.u . . . 4  |-  ( ph  ->  U  e.  V )
21 elex 3022 . . . 4  |-  ( U  e.  V  ->  U  e.  _V )
2220, 21syl 17 . . 3  |-  ( ph  ->  U  e.  _V )
23 tpex 6578 . . . 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 5938 . 2  |-  ( ph  ->  (ExtStrCat `  U )  =  { <. ( Base `  ndx ) ,  U >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
261, 25syl5eq 2498 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 1448    e. wcel 1891   _Vcvv 3013   {ctp 3940   <.cop 3942    |-> cmpt 4433    X. cxp 4810    o. ccom 4816   ` cfv 5561  (class class class)co 6276    |-> cmpt2 6278   1stc1st 6779   2ndc2nd 6780    ^m cmap 7459   ndxcnx 15129   Basecbs 15132   Hom chom 15212  compcco 15213  ExtStrCatcestrc 16018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-iota 5525  df-fun 5563  df-fv 5569  df-oprab 6280  df-mpt2 6281  df-estrc 16019
This theorem is referenced by:  estrcbas  16021  estrchomfval  16022  estrccofval  16025  dfrngc2  40299  dfringc2  40345
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