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

Theorem setcco 14193
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 14192 . . 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 734 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
6 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  v  =  <. X ,  Y >. )
76fveq2d 5691 . . . . . 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 6319 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  U )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
118, 9, 10syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1211adantr 452 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
137, 12eqtrd 2436 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
145, 13oveq12d 6058 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
z  ^m  ( 2nd `  v ) )  =  ( Z  ^m  Y
) )
156fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  ( 1st `  <. X ,  Y >. )
)
16 op1stg 6318 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  U )  ->  ( 1st `  <. X ,  Y >. )  =  X )
178, 9, 16syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
1817adantr 452 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
1915, 18eqtrd 2436 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  X )
2013, 19oveq12d 6058 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  ^m  ( 1st `  v ) )  =  ( Y  ^m  X
) )
21 eqidd 2405 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.  f )  =  ( g  o.  f ) )
2214, 20, 21mpt2eq123dv 6095 . . 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 4869 . . . 4  |-  ( ( X  e.  U  /\  Y  e.  U )  -> 
<. X ,  Y >.  e.  ( U  X.  U
) )
248, 9, 23syl2anc 643 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( U  X.  U
) )
25 setcco.z . . 3  |-  ( ph  ->  Z  e.  U )
26 ovex 6065 . . . . 5  |-  ( Z  ^m  Y )  e. 
_V
27 ovex 6065 . . . . 5  |-  ( Y  ^m  X )  e. 
_V
2826, 27mpt2ex 6384 . . . 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 6160 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X ) 
|->  ( g  o.  f
) ) )
31 simprl 733 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
32 simprr 734 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
3331, 32coeq12d 4996 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g  o.  f
)  =  ( G  o.  F ) )
34 setcco.g . . 3  |-  ( ph  ->  G : Y --> Z )
35 elmapg 6990 . . . 4  |-  ( ( Z  e.  U  /\  Y  e.  U )  ->  ( G  e.  ( Z  ^m  Y )  <-> 
G : Y --> Z ) )
3625, 9, 35syl2anc 643 . . 3  |-  ( ph  ->  ( G  e.  ( Z  ^m  Y )  <-> 
G : Y --> Z ) )
3734, 36mpbird 224 . 2  |-  ( ph  ->  G  e.  ( Z  ^m  Y ) )
38 setcco.f . . 3  |-  ( ph  ->  F : X --> Y )
39 elmapg 6990 . . . 4  |-  ( ( Y  e.  U  /\  X  e.  U )  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
409, 8, 39syl2anc 643 . . 3  |-  ( ph  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
4138, 40mpbird 224 . 2  |-  ( ph  ->  F  e.  ( Y  ^m  X ) )
42 coexg 5371 . . 3  |-  ( ( G  e.  ( Z  ^m  Y )  /\  F  e.  ( Y  ^m  X ) )  -> 
( G  o.  F
)  e.  _V )
4337, 41, 42syl2anc 643 . 2  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
4430, 33, 37, 41, 43ovmpt2d 6160 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777    X. cxp 4835    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307    ^m cmap 6977  compcco 13496   SetCatcsetc 14185
This theorem is referenced by:  setccatid  14194  setcmon  14197  setcepi  14198  setcsect  14199  resssetc  14202  hofcllem  14310  yonedalem4c  14329  yonedalem3b  14331  yonedainv  14333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-setc 14186
  Copyright terms: Public domain W3C validator