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Theorem setcco 13931
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 13930 . . 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 733 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  z  =  Z )
6 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  v  =  <. X ,  Y >. )
76fveq2d 5545 . . . . . 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 6149 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  U )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
118, 9, 10syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
1211adantr 451 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
137, 12eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
145, 13oveq12d 5892 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
z  ^m  ( 2nd `  v ) )  =  ( Z  ^m  Y
) )
156fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  ( 1st `  <. X ,  Y >. )
)
16 op1stg 6148 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  U )  ->  ( 1st `  <. X ,  Y >. )  =  X )
178, 9, 16syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
1817adantr 451 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
1915, 18eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  X )
2013, 19oveq12d 5892 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  ^m  ( 1st `  v ) )  =  ( Y  ^m  X
) )
21 eqidd 2297 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.  f )  =  ( g  o.  f ) )
2214, 20, 21mpt2eq123dv 5926 . . 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 4737 . . . 4  |-  ( ( X  e.  U  /\  Y  e.  U )  -> 
<. X ,  Y >.  e.  ( U  X.  U
) )
248, 9, 23syl2anc 642 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( U  X.  U
) )
25 setcco.z . . 3  |-  ( ph  ->  Z  e.  U )
26 ovex 5899 . . . . 5  |-  ( Z  ^m  Y )  e. 
_V
27 ovex 5899 . . . . 5  |-  ( Y  ^m  X )  e. 
_V
2826, 27mpt2ex 6214 . . . 4  |-  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X )  |->  ( g  o.  f ) )  e.  _V
2928a1i 10 . . 3  |-  ( ph  ->  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X ) 
|->  ( g  o.  f
) )  e.  _V )
304, 22, 24, 25, 29ovmpt2d 5991 . 2  |-  ( ph  ->  ( <. X ,  Y >.  .x.  Z )  =  ( g  e.  ( Z  ^m  Y ) ,  f  e.  ( Y  ^m  X ) 
|->  ( g  o.  f
) ) )
31 simprl 732 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
32 simprr 733 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
3331, 32coeq12d 4864 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g  o.  f
)  =  ( G  o.  F ) )
34 setcco.g . . 3  |-  ( ph  ->  G : Y --> Z )
35 elmapg 6801 . . . 4  |-  ( ( Z  e.  U  /\  Y  e.  U )  ->  ( G  e.  ( Z  ^m  Y )  <-> 
G : Y --> Z ) )
3625, 9, 35syl2anc 642 . . 3  |-  ( ph  ->  ( G  e.  ( Z  ^m  Y )  <-> 
G : Y --> Z ) )
3734, 36mpbird 223 . 2  |-  ( ph  ->  G  e.  ( Z  ^m  Y ) )
38 setcco.f . . 3  |-  ( ph  ->  F : X --> Y )
39 elmapg 6801 . . . 4  |-  ( ( Y  e.  U  /\  X  e.  U )  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
409, 8, 39syl2anc 642 . . 3  |-  ( ph  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
4138, 40mpbird 223 . 2  |-  ( ph  ->  F  e.  ( Y  ^m  X ) )
42 coexg 5231 . . 3  |-  ( ( G  e.  ( Z  ^m  Y )  /\  F  e.  ( Y  ^m  X ) )  -> 
( G  o.  F
)  e.  _V )
4337, 41, 42syl2anc 642 . 2  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
4430, 33, 37, 41, 43ovmpt2d 5991 1  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  =  ( G  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788  compcco 13236   SetCatcsetc 13923
This theorem is referenced by:  setccatid  13932  setcmon  13935  setcepi  13936  setcsect  13937  resssetc  13940  hofcllem  14048  yonedalem4c  14067  yonedalem3b  14069  yonedainv  14071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-setc 13924
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