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

Theorem setcco 15264
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 )
setcco.x  |-  ( ph  ->  X  e.  U )
setcco.y  |-  ( ph  ->  Y  e.  U )
setcco.z  |-  ( ph  ->  Z  e.  U )
setcco.f  |-  ( ph  ->  F : X --> Y )
setcco.g  |-  ( ph  ->  G : Y --> Z )
Assertion
Ref Expression
setcco  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.  F ) )

Proof of Theorem setcco
Dummy variables  f 
g  v  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setcbas.c . . . 4  |-  C  =  ( SetCat `  U )
2 setcbas.u . . . 4  |-  ( ph  ->  U  e.  V )
3 setcco.o . . . 4  |-  .x.  =  (comp `  C )
41, 2, 3setccofval 15263 . . 3  |-  ( 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 ) ) ) )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
6 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  v  =  <. X ,  Y >. )
76fveq2d 5868 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  ( 2nd `  <. X ,  Y >. )
)
8 setcco.x . . . . . . . 8  |-  ( ph  ->  X  e.  U )
9 setcco.y . . . . . . . 8  |-  ( ph  ->  Y  e.  U )
10 op2ndg 6794 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  U )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
118, 9, 10syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
137, 12eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
145, 13oveq12d 6300 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
z  ^m  ( 2nd `  v ) )  =  ( Z  ^m  Y
) )
156fveq2d 5868 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  ( 1st `  <. X ,  Y >. )
)
16 op1stg 6793 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  U )  ->  ( 1st `  <. X ,  Y >. )  =  X )
178, 9, 16syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
1817adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
1915, 18eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  X )
2013, 19oveq12d 6300 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  ^m  ( 1st `  v ) )  =  ( Y  ^m  X
) )
21 eqidd 2468 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.  f )  =  ( g  o.  f ) )
2214, 20, 21mpt2eq123dv 6341 . . 3  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  e.  ( z  ^m  ( 2nd `  v
) ) ,  f  e.  ( ( 2nd `  v )  ^m  ( 1st `  v ) ) 
|->  ( g  o.  f
) )  =  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X )  |->  ( g  o.  f ) ) )
23 opelxpi 5030 . . . 4  |-  ( ( X  e.  U  /\  Y  e.  U )  -> 
<. X ,  Y >.  e.  ( U  X.  U
) )
248, 9, 23syl2anc 661 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( U  X.  U
) )
25 setcco.z . . 3  |-  ( ph  ->  Z  e.  U )
26 ovex 6307 . . . . 5  |-  ( Z  ^m  Y )  e. 
_V
27 ovex 6307 . . . . 5  |-  ( Y  ^m  X )  e. 
_V
2826, 27mpt2ex 6857 . . . 4  |-  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X )  |->  ( g  o.  f ) )  e.  _V
2928a1i 11 . . 3  |-  ( ph  ->  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X ) 
|->  ( g  o.  f
) )  e.  _V )
304, 22, 24, 25, 29ovmpt2d 6412 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X ) 
|->  ( g  o.  f
) ) )
31 simprl 755 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
32 simprr 756 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
3331, 32coeq12d 5165 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g  o.  f
)  =  ( G  o.  F ) )
34 setcco.g . . 3  |-  ( ph  ->  G : Y --> Z )
35 elmapg 7430 . . . 4  |-  ( ( Z  e.  U  /\  Y  e.  U )  ->  ( G  e.  ( Z  ^m  Y )  <-> 
G : Y --> Z ) )
3625, 9, 35syl2anc 661 . . 3  |-  ( ph  ->  ( G  e.  ( Z  ^m  Y )  <-> 
G : Y --> Z ) )
3734, 36mpbird 232 . 2  |-  ( ph  ->  G  e.  ( Z  ^m  Y ) )
38 setcco.f . . 3  |-  ( ph  ->  F : X --> Y )
39 elmapg 7430 . . . 4  |-  ( ( Y  e.  U  /\  X  e.  U )  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
409, 8, 39syl2anc 661 . . 3  |-  ( ph  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
4138, 40mpbird 232 . 2  |-  ( ph  ->  F  e.  ( Y  ^m  X ) )
42 coexg 6732 . . 3  |-  ( ( G  e.  ( Z  ^m  Y )  /\  F  e.  ( Y  ^m  X ) )  -> 
( G  o.  F
)  e.  _V )
4337, 41, 42syl2anc 661 . 2  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
4430, 33, 37, 41, 43ovmpt2d 6412 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033    X. cxp 4997    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780    ^m cmap 7417  compcco 14563   SetCatcsetc 15256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-hom 14575  df-cco 14576  df-setc 15257
This theorem is referenced by:  setccatid  15265  setcmon  15268  setcepi  15269  setcsect  15270  resssetc  15273  hofcllem  15381  yonedalem4c  15400  yonedalem3b  15402  yonedainv  15404
  Copyright terms: Public domain W3C validator