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Theorem setcco 14238
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 14237 . . 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 5732 . . . . . 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 6360 . . . . . . . 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 2468 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 2nd `  v )  =  Y )
145, 13oveq12d 6099 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
z  ^m  ( 2nd `  v ) )  =  ( Z  ^m  Y
) )
156fveq2d 5732 . . . . . 6  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  ( 1st `  <. X ,  Y >. )
)
16 op1stg 6359 . . . . . . . 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 2468 . . . . 5  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  ( 1st `  v )  =  X )
2013, 19oveq12d 6099 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
( 2nd `  v
)  ^m  ( 1st `  v ) )  =  ( Y  ^m  X
) )
21 eqidd 2437 . . . 4  |-  ( (
ph  /\  ( v  =  <. X ,  Y >.  /\  z  =  Z ) )  ->  (
g  o.  f )  =  ( g  o.  f ) )
2214, 20, 21mpt2eq123dv 6136 . . 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 4910 . . . 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 6106 . . . . 5  |-  ( Z  ^m  Y )  e. 
_V
27 ovex 6106 . . . . 5  |-  ( Y  ^m  X )  e. 
_V
2826, 27mpt2ex 6425 . . . 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 6201 . 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 5037 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g  o.  f
)  =  ( G  o.  F ) )
34 setcco.g . . 3  |-  ( ph  ->  G : Y --> Z )
35 elmapg 7031 . . . 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 7031 . . . 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 5412 . . 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 6201 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 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348    ^m cmap 7018  compcco 13541   SetCatcsetc 14230
This theorem is referenced by:  setccatid  14239  setcmon  14242  setcepi  14243  setcsect  14244  resssetc  14247  hofcllem  14355  yonedalem4c  14374  yonedalem3b  14376  yonedainv  14378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-hom 13553  df-cco 13554  df-setc 14231
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