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

Theorem setccofval 14124
Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcbas.c  |-  C  =  ( SetCat `  U )
setcbas.u  |-  ( ph  ->  U  e.  V )
setcco.o  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
setccofval  |-  ( 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 ) ) ) )
Distinct variable groups:    f, g,
v, z, ph    v, U, z
Allowed substitution hints:    C( z, v, f, g)    .x. ( z, v, f, g)    U( f, g)    V( z, v, f, g)

Proof of Theorem setccofval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setcco.o . 2  |-  .x.  =  (comp `  C )
2 setcbas.c . . . . 5  |-  C  =  ( SetCat `  U )
3 setcbas.u . . . . 5  |-  ( ph  ->  U  e.  V )
4 eqid 2366 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
52, 3, 4setchomfval 14121 . . . . 5  |-  ( ph  ->  (  Hom  `  C
)  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
6 eqidd 2367 . . . . 5  |-  ( ph  ->  ( 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 ) ) ) )
72, 3, 5, 6setcval 14119 . . . 4  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  C
) >. ,  <. (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
) ) ) >. } )
87fveq2d 5636 . . 3  |-  ( ph  ->  (comp `  C )  =  (comp `  { <. ( Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  C
) >. ,  <. (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
) ) ) >. } ) )
9 xpexg 4903 . . . . . 6  |-  ( ( U  e.  V  /\  U  e.  V )  ->  ( U  X.  U
)  e.  _V )
103, 3, 9syl2anc 642 . . . . 5  |-  ( ph  ->  ( U  X.  U
)  e.  _V )
11 eqid 2366 . . . . . 6  |-  ( 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
) ) )
1211mpt2exg 6323 . . . . 5  |-  ( ( ( U  X.  U
)  e.  _V  /\  U  e.  V )  ->  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v
) )  |->  ( g  o.  f ) ) )  e.  _V )
1310, 3, 12syl2anc 642 . . . 4  |-  ( ph  ->  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v
) )  |->  ( g  o.  f ) ) )  e.  _V )
14 catstr 14041 . . . . 5  |-  { <. (
Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  C
) >. ,  <. (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
) ) ) >. } Struct  <. 1 , ; 1 5 >.
15 df-cco 13441 . . . . . 6  |- comp  = Slot ; 1 5
16 1nn0 10130 . . . . . . 7  |-  1  e.  NN0
17 5nn 10029 . . . . . . 7  |-  5  e.  NN
1816, 17decnncl 10288 . . . . . 6  |- ; 1 5  e.  NN
1915, 18ndxid 13377 . . . . 5  |- comp  = Slot  (comp ` 
ndx )
20 snsstp3 3866 . . . . 5  |-  { <. (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
) ) ) >. }  C_  { <. ( Base `  ndx ) ,  U >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  C
) >. ,  <. (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
) ) ) >. }
2114, 19, 20strfv 13388 . . . 4  |-  ( ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) ) )  e. 
_V  ->  ( 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 `  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  (  Hom  `  C ) >. ,  <. (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
) ) ) >. } ) )
2213, 21syl 15 . . 3  |-  ( ph  ->  ( 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 `  { <. ( Base `  ndx ) ,  U >. , 
<. (  Hom  `  ndx ) ,  (  Hom  `  C ) >. ,  <. (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
) ) ) >. } ) )
238, 22eqtr4d 2401 . 2  |-  ( ph  ->  (comp `  C )  =  ( v  e.  ( U  X.  U
) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  ( ( 2nd `  v
)  ^m  ( 1st `  v ) )  |->  ( g  o.  f ) ) ) )
241, 23syl5eq 2410 1  |-  ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   _Vcvv 2873   {ctp 3731   <.cop 3732    X. cxp 4790    o. ccom 4796   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   1stc1st 6247   2ndc2nd 6248    ^m cmap 6915   1c1 8885   5c5 9945  ;cdc 10275   ndxcnx 13353   Basecbs 13356    Hom chom 13427  compcco 13428   SetCatcsetc 14117
This theorem is referenced by:  setcco  14125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-hom 13440  df-cco 13441  df-setc 14118
  Copyright terms: Public domain W3C validator