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Theorem 2oppccomf 13644
Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13656. (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 2296 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
31, 2oppcbas 13637 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  O )
4 eqid 2296 . . . . . . . 8  |-  (comp `  O )  =  (comp `  O )
5 eqid 2296 . . . . . . . 8  |-  (oppCat `  O )  =  (oppCat `  O )
6 simpr1 961 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  x  e.  ( Base `  C )
)
7 simpr2 962 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  y  e.  ( Base `  C )
)
8 simpr3 963 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  z  e.  ( Base `  C )
)
93, 4, 5, 6, 7, 8oppcco 13636 . . . . . . 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 2296 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
112, 10, 1, 8, 7, 6oppcco 13636 . . . . . . 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 2329 . . . . . 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 2640 . . . . 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 2640 . . . 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 2649 . . 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 2296 . . . 4  |-  (comp `  (oppCat `  O ) )  =  (comp `  (oppCat `  O ) )
17 eqid 2296 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
18 eqidd 2297 . . . 4  |-  (  T. 
->  ( Base `  C
)  =  ( Base `  C ) )
191, 22oppcbas 13642 . . . . 5  |-  ( Base `  C )  =  (
Base `  (oppCat `  O
) )
2019a1i 10 . . . 4  |-  (  T. 
->  ( Base `  C
)  =  ( Base `  (oppCat `  O )
) )
2112oppchomf 13643 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  (oppCat `  O )
)
2221a1i 10 . . . 4  |-  (  T. 
->  (  Homf 
`  C )  =  (  Homf 
`  (oppCat `  O )
) )
2310, 16, 17, 18, 20, 22comfeq 13625 . . 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 223 . 2  |-  (  T. 
->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
2524trud 1314 1  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236    Homf chomf 13584  compfccomf 13585  oppCatcoppc 13630
This theorem is referenced by:  oppcepi  13658  oppchofcl  14050  oppcyon  14059  oyoncl  14060
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-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-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-hom 13248  df-cco 13249  df-homf 13588  df-comf 13589  df-oppc 13631
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