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Theorem oppccomfpropd 15142
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 2457 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2457 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2457 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
4 eqid 2457 . . . . . 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 1040 . . . . . 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 1039 . . . . . 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 1038 . . . . . 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 757 . . . . . . 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 2457 . . . . . . . 8  |-  (oppCat `  C )  =  (oppCat `  C )
142, 13oppchom 15130 . . . . . . 7  |-  ( y ( Hom  `  (oppCat `  C ) ) z )  =  ( z ( Hom  `  C
) y )
1512, 14syl6eleq 2555 . . . . . 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 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 ) ) )  ->  f  e.  ( x ( Hom  `  (oppCat `  C )
) y ) )
172, 13oppchom 15130 . . . . . . 7  |-  ( x ( Hom  `  (oppCat `  C ) ) y )  =  ( y ( Hom  `  C
) x )
1816, 17syl6eleq 2555 . . . . . 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 15123 . . . . 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 15132 . . . . 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 2457 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
22 eqid 2457 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
235homfeqbas 15111 . . . . . . . 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 2547 . . . . . 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 2547 . . . . . 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 2547 . . . . . 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 15132 . . . . 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 2508 . . . 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 2878 . . 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 2879 . 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 2457 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
33 eqid 2457 . . 3  |-  (comp `  (oppCat `  D ) )  =  (comp `  (oppCat `  D ) )
34 eqid 2457 . . 3  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
3513, 1oppcbas 15133 . . . 4  |-  ( Base `  C )  =  (
Base `  (oppCat `  C
) )
3635a1i 11 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  C )
) )
3722, 21oppcbas 15133 . . . 4  |-  ( Base `  D )  =  (
Base `  (oppCat `  D
) )
3823, 37syl6eq 2514 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  D )
) )
395oppchomfpropd 15141 . . 3  |-  ( ph  ->  ( Hom f  `  (oppCat `  C
) )  =  ( Hom f  `  (oppCat `  D )
) )
4032, 33, 34, 36, 38, 39comfeq 15121 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   <.cop 4038   ` cfv 5594  (class class class)co 6296   Basecbs 14643   Hom chom 14722  compcco 14723   Hom f chomf 15082  compfccomf 15083  oppCatcoppc 15126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-hom 14735  df-cco 14736  df-homf 15086  df-comf 15087  df-oppc 15127
This theorem is referenced by:  yonpropd  15663
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