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Theorem oppccomfpropd 14671
Description: If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
oppchomfpropd.1  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
oppccomfpropd.1  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
Assertion
Ref Expression
oppccomfpropd  |-  ( ph  ->  (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) ) )

Proof of Theorem oppccomfpropd
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2443 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2443 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
4 eqid 2443 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 oppchomfpropd.1 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
65ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( Hom f  `  C
)  =  ( Hom f  `  D ) )
7 oppccomfpropd.1 . . . . . . 7  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
87ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  (compf `  C )  =  (compf `  D ) )
9 simplr3 1032 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  z  e.  ( Base `  C )
)
10 simplr2 1031 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  y  e.  ( Base `  C )
)
11 simplr1 1030 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  x  e.  ( Base `  C )
)
12 simprr 756 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) )
13 eqid 2443 . . . . . . . 8  |-  (oppCat `  C )  =  (oppCat `  C )
142, 13oppchom 14659 . . . . . . 7  |-  ( y ( Hom  `  (oppCat `  C ) ) z )  =  ( z ( Hom  `  C
) y )
1512, 14syl6eleq 2533 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  g  e.  ( z ( Hom  `  C ) y ) )
16 simprl 755 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) )
172, 13oppchom 14659 . . . . . . 7  |-  ( x ( Hom  `  (oppCat `  C ) ) y )  =  ( y ( Hom  `  C
) x )
1816, 17syl6eleq 2533 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  f  e.  ( y ( Hom  `  C ) x ) )
191, 2, 3, 4, 6, 8, 9, 10, 11, 15, 18comfeqval 14652 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( f
( <. z ,  y
>. (comp `  C )
x ) g )  =  ( f (
<. z ,  y >.
(comp `  D )
x ) g ) )
201, 3, 13, 11, 10, 9oppcco 14661 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  C
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  C ) x ) g ) )
21 eqid 2443 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
22 eqid 2443 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
235homfeqbas 14640 . . . . . . . 8  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
2423ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( Base `  C )  =  (
Base `  D )
)
2511, 24eleqtrd 2519 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  x  e.  ( Base `  D )
)
2610, 24eleqtrd 2519 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  y  e.  ( Base `  D )
)
279, 24eleqtrd 2519 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  z  e.  ( Base `  D )
)
2821, 4, 22, 25, 26, 27oppcco 14661 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  D
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  D ) x ) g ) )
2919, 20, 283eqtr4d 2485 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  z  e.  ( Base `  C )
) )  /\  (
f  e.  ( x ( Hom  `  (oppCat `  C ) ) y )  /\  g  e.  ( y ( Hom  `  (oppCat `  C )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) )
3029ralrimivva 2813 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) ) )  ->  A. f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) A. g  e.  ( y
( Hom  `  (oppCat `  C ) ) z ) ( g (
<. x ,  y >.
(comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) )
3130ralrimivvva 2814 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) A. g  e.  ( y
( Hom  `  (oppCat `  C ) ) z ) ( g (
<. x ,  y >.
(comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) )
32 eqid 2443 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
33 eqid 2443 . . 3  |-  (comp `  (oppCat `  D ) )  =  (comp `  (oppCat `  D ) )
34 eqid 2443 . . 3  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
3513, 1oppcbas 14662 . . . 4  |-  ( Base `  C )  =  (
Base `  (oppCat `  C
) )
3635a1i 11 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  C )
) )
3722, 21oppcbas 14662 . . . 4  |-  ( Base `  D )  =  (
Base `  (oppCat `  D
) )
3823, 37syl6eq 2491 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  D )
) )
395oppchomfpropd 14670 . . 3  |-  ( ph  ->  ( Hom f  `  (oppCat `  C
) )  =  ( Hom f  `  (oppCat `  D )
) )
4032, 33, 34, 36, 38, 39comfeq 14650 . 2  |-  ( ph  ->  ( (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) )  <->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. z  e.  ( Base `  C
) A. f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) A. g  e.  ( y
( Hom  `  (oppCat `  C ) ) z ) ( g (
<. x ,  y >.
(comp `  (oppCat `  C
) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  (oppCat `  D )
) z ) f ) ) )
4131, 40mpbird 232 1  |-  ( ph  ->  (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   <.cop 3888   ` cfv 5423  (class class class)co 6096   Basecbs 14179   Hom chom 14254  compcco 14255   Hom f chomf 14609  compfccomf 14610  oppCatcoppc 14655
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-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-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  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-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-hom 14267  df-cco 14268  df-homf 14613  df-comf 14614  df-oppc 14656
This theorem is referenced by:  yonpropd  15083
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