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Theorem 2oppccomf 15102
Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 15114. (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 15095 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  O )
4 eqid 2443 . . . . . . . 8  |-  (comp `  O )  =  (comp `  O )
5 eqid 2443 . . . . . . . 8  |-  (oppCat `  O )  =  (oppCat `  O )
6 simpr1 1003 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  x  e.  ( Base `  C )
)
7 simpr2 1004 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  y  e.  ( Base `  C )
)
8 simpr3 1005 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  z  e.  ( Base `  C )
)
93, 4, 5, 6, 7, 8oppcco 15094 . . . . . . 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 15094 . . . . . . 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 2485 . . . . . 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 2858 . . . . 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 2858 . . . 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 2865 . . 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 15100 . . . . 5  |-  ( Base `  C )  =  (
Base `  (oppCat `  O
) )
2019a1i 11 . . . 4  |-  ( T. 
->  ( Base `  C
)  =  ( Base `  (oppCat `  O )
) )
2112oppchomf 15101 . . . . 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 15083 . . 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 1392 1  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 974    = wceq 1383   T. wtru 1384    e. wcel 1804   A.wral 2793   <.cop 4020   ` cfv 5578  (class class class)co 6281   Basecbs 14614   Hom chom 14690  compcco 14691   Hom f chomf 15045  compfccomf 15046  oppCatcoppc 15088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-hom 14703  df-cco 14704  df-homf 15049  df-comf 15050  df-oppc 15089
This theorem is referenced by:  oppcepi  15116  oppchofcl  15508  oppcyon  15517  oyoncl  15518
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