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Theorem oppccomfpropd 13630
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  ->  (  Homf 
`  C )  =  (  Homf 
`  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 2283 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2283 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2283 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
4 eqid 2283 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 oppchomfpropd.1 . . . . . . 7  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
65ad2antrr 706 . . . . . 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 ) ) )  ->  (  Homf  `  C
)  =  (  Homf  `  D ) )
7 oppccomfpropd.1 . . . . . . 7  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
87ad2antrr 706 . . . . . 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 999 . . . . . 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 998 . . . . . 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 997 . . . . . 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 733 . . . . . . 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 2283 . . . . . . . 8  |-  (oppCat `  C )  =  (oppCat `  C )
142, 13oppchom 13618 . . . . . . 7  |-  ( y (  Hom  `  (oppCat `  C ) ) z )  =  ( z (  Hom  `  C
) y )
1512, 14syl6eleq 2373 . . . . . 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 732 . . . . . . 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 13618 . . . . . . 7  |-  ( x (  Hom  `  (oppCat `  C ) ) y )  =  ( y (  Hom  `  C
) x )
1816, 17syl6eleq 2373 . . . . . 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 13611 . . . . 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 13620 . . . . 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 2283 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
22 eqid 2283 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
235homfeqbas 13599 . . . . . . . 8  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
2423ad2antrr 706 . . . . . . 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 2359 . . . . . 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 2359 . . . . . 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 2359 . . . . . 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 13620 . . . . 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 2325 . . . 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 2635 . . 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 2636 . 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 2283 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
33 eqid 2283 . . 3  |-  (comp `  (oppCat `  D ) )  =  (comp `  (oppCat `  D ) )
34 eqid 2283 . . 3  |-  (  Hom  `  (oppCat `  C )
)  =  (  Hom  `  (oppCat `  C )
)
3513, 1oppcbas 13621 . . . 4  |-  ( Base `  C )  =  (
Base `  (oppCat `  C
) )
3635a1i 10 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  C )
) )
3722, 21oppcbas 13621 . . . 4  |-  ( Base `  D )  =  (
Base `  (oppCat `  D
) )
3823, 37syl6eq 2331 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  (oppCat `  D )
) )
395oppchomfpropd 13629 . . 3  |-  ( ph  ->  (  Homf 
`  (oppCat `  C )
)  =  (  Homf  `  (oppCat `  D ) ) )
4032, 33, 34, 36, 38, 39comfeq 13609 . 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 223 1  |-  ( ph  ->  (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220    Homf chomf 13568  compfccomf 13569  oppCatcoppc 13614
This theorem is referenced by:  yonpropd  14042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-hom 13232  df-cco 13233  df-homf 13572  df-comf 13573  df-oppc 13615
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