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Theorem 2oppccomf 14664
Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 14676. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypothesis
Ref Expression
oppcbas.1  |-  O  =  (oppCat `  C )
Assertion
Ref Expression
2oppccomf  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )

Proof of Theorem 2oppccomf
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppcbas.1 . . . . . . . . 9  |-  O  =  (oppCat `  C )
2 eqid 2443 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
31, 2oppcbas 14657 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  O )
4 eqid 2443 . . . . . . . 8  |-  (comp `  O )  =  (comp `  O )
5 eqid 2443 . . . . . . . 8  |-  (oppCat `  O )  =  (oppCat `  O )
6 simpr1 994 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  x  e.  ( Base `  C )
)
7 simpr2 995 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  y  e.  ( Base `  C )
)
8 simpr3 996 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  z  e.  ( Base `  C )
)
93, 4, 5, 6, 7, 8oppcco 14656 . . . . . . 7  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  O
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  O ) x ) g ) )
10 eqid 2443 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
112, 10, 1, 8, 7, 6oppcco 14656 . . . . . . 7  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  ( f
( <. z ,  y
>. (comp `  O )
x ) g )  =  ( g (
<. x ,  y >.
(comp `  C )
z ) f ) )
129, 11eqtr2d 2476 . . . . . 6  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  (oppCat `  O
) ) z ) f ) )
1312ralrimivw 2800 . . . . 5  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  A. g  e.  ( y ( Hom  `  C ) z ) ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  (oppCat `  O
) ) z ) f ) )
1413ralrimivw 2800 . . . 4  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  A. f  e.  ( x ( Hom  `  C ) y ) A. g  e.  ( y ( Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  O )
) z ) f ) )
1514ralrimivvva 2809 . . 3  |-  ( T. 
->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x ( Hom  `  C ) y ) A. g  e.  ( y ( Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  O )
) z ) f ) )
16 eqid 2443 . . . 4  |-  (comp `  (oppCat `  O ) )  =  (comp `  (oppCat `  O ) )
17 eqid 2443 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
18 eqidd 2444 . . . 4  |-  ( T. 
->  ( Base `  C
)  =  ( Base `  C ) )
191, 22oppcbas 14662 . . . . 5  |-  ( Base `  C )  =  (
Base `  (oppCat `  O
) )
2019a1i 11 . . . 4  |-  ( T. 
->  ( Base `  C
)  =  ( Base `  (oppCat `  O )
) )
2112oppchomf 14663 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2221a1i 11 . . . 4  |-  ( T. 
->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2310, 16, 17, 18, 20, 22comfeq 14645 . . 3  |-  ( T. 
->  ( (compf `  C )  =  (compf `  (oppCat `  O ) )  <->  A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C
) A. f  e.  ( x ( Hom  `  C ) y ) A. g  e.  ( y ( Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  O )
) z ) f ) ) )
2415, 23mpbird 232 . 2  |-  ( T. 
->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
2524trud 1378 1  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1369   T. wtru 1370    e. wcel 1756   A.wral 2715   <.cop 3883   ` cfv 5418  (class class class)co 6091   Basecbs 14174   Hom chom 14249  compcco 14250   Hom f chomf 14604  compfccomf 14605  oppCatcoppc 14650
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-hom 14262  df-cco 14263  df-homf 14608  df-comf 14609  df-oppc 14651
This theorem is referenced by:  oppcepi  14678  oppchofcl  15070  oppcyon  15079  oyoncl  15080
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