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Theorem oppccomfpropd 14662
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 2441 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2441 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2441 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
4 eqid 2441 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 oppchomfpropd.1 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
65ad2antrr 720 . . . . . 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 720 . . . . . 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 1027 . . . . . 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 1026 . . . . . 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 1025 . . . . . 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 751 . . . . . . 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 2441 . . . . . . . 8  |-  (oppCat `  C )  =  (oppCat `  C )
142, 13oppchom 14650 . . . . . . 7  |-  ( y ( Hom  `  (oppCat `  C ) ) z )  =  ( z ( Hom  `  C
) y )
1512, 14syl6eleq 2531 . . . . . 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 750 . . . . . . 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 14650 . . . . . . 7  |-  ( x ( Hom  `  (oppCat `  C ) ) y )  =  ( y ( Hom  `  C
) x )
1816, 17syl6eleq 2531 . . . . . 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 14643 . . . . 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 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 ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  C
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  C ) x ) g ) )
21 eqid 2441 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
22 eqid 2441 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
235homfeqbas 14631 . . . . . . . 8  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
2423ad2antrr 720 . . . . . . 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 2517 . . . . . 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 2517 . . . . . 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 2517 . . . . . 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 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 ) ) )  ->  ( g
( <. x ,  y
>. (comp `  (oppCat `  D
) ) z ) f )  =  ( f ( <. z ,  y >. (comp `  D ) x ) g ) )
2919, 20, 283eqtr4d 2483 . . . 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 2806 . . 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 2807 . 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 2441 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
33 eqid 2441 . . 3  |-  (comp `  (oppCat `  D ) )  =  (comp `  (oppCat `  D ) )
34 eqid 2441 . . 3  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
3513, 1oppcbas 14653 . . . 4  |-  ( Base `  C )  =  (
Base `  (oppCat `  C
) )
3635a1i 11 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  C )
) )
3722, 21oppcbas 14653 . . . 4  |-  ( Base `  D )  =  (
Base `  (oppCat `  D
) )
3823, 37syl6eq 2489 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  D )
) )
395oppchomfpropd 14661 . . 3  |-  ( ph  ->  ( Hom f  `  (oppCat `  C
) )  =  ( Hom f  `  (oppCat `  D )
) )
4032, 33, 34, 36, 38, 39comfeq 14641 . 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   <.cop 3880   ` cfv 5415  (class class class)co 6090   Basecbs 14170   Hom chom 14245  compcco 14246   Hom f chomf 14600  compfccomf 14601  oppCatcoppc 14646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-hom 14258  df-cco 14259  df-homf 14604  df-comf 14605  df-oppc 14647
This theorem is referenced by:  yonpropd  15074
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