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Theorem setcco 14956
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 14955 . . 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 5700 . . . . . 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 6595 . . . . . . . 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 2475 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
145, 13oveq12d 6114 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
z  ^m  ( 2nd `  v ) )  =  ( Z  ^m  Y
) )
156fveq2d 5700 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  ( 1st `  <. X ,  Y >. )
)
16 op1stg 6594 . . . . . . . 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 2475 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  X )
2013, 19oveq12d 6114 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  ^m  ( 1st `  v ) )  =  ( Y  ^m  X
) )
21 eqidd 2444 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.  f )  =  ( g  o.  f ) )
2214, 20, 21mpt2eq123dv 6153 . . 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 4876 . . . 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 6121 . . . . 5  |-  ( Z  ^m  Y )  e. 
_V
27 ovex 6121 . . . . 5  |-  ( Y  ^m  X )  e. 
_V
2826, 27mpt2ex 6655 . . . 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 6223 . 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 5009 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g  o.  f
)  =  ( G  o.  F ) )
34 setcco.g . . 3  |-  ( ph  ->  G : Y --> Z )
35 elmapg 7232 . . . 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 7232 . . . 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 6533 . . 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 6223 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 1369    e. wcel 1756   _Vcvv 2977   <.cop 3888    X. cxp 4843    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581    ^m cmap 7219  compcco 14255   SetCatcsetc 14948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-hom 14267  df-cco 14268  df-setc 14949
This theorem is referenced by:  setccatid  14957  setcmon  14960  setcepi  14961  setcsect  14962  resssetc  14965  hofcllem  15073  yonedalem4c  15092  yonedalem3b  15094  yonedainv  15096
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