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Theorem catcoppccl 14997
Description: The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcoppccl.c  |-  C  =  (CatCat `  U )
catcoppccl.b  |-  B  =  ( Base `  C
)
catcoppccl.o  |-  O  =  (oppCat `  X )
catcoppccl.1  |-  ( ph  ->  U  e. WUni )
catcoppccl.2  |-  ( ph  ->  om  e.  U )
catcoppccl.3  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
catcoppccl  |-  ( ph  ->  O  e.  B )

Proof of Theorem catcoppccl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcoppccl.3 . . . . 5  |-  ( ph  ->  X  e.  B )
2 eqid 2443 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2443 . . . . . 6  |-  ( Hom  `  X )  =  ( Hom  `  X )
4 eqid 2443 . . . . . 6  |-  (comp `  X )  =  (comp `  X )
5 catcoppccl.o . . . . . 6  |-  O  =  (oppCat `  X )
62, 3, 4, 5oppcval 14673 . . . . 5  |-  ( X  e.  B  ->  O  =  ( ( X sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  X ) >. ) sSet  <.
(comp `  ndx ) ,  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) )
>. ) )
71, 6syl 16 . . . 4  |-  ( ph  ->  O  =  ( ( X sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  X ) >. ) sSet  <.
(comp `  ndx ) ,  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) )
>. ) )
8 catcoppccl.1 . . . . 5  |-  ( ph  ->  U  e. WUni )
9 inss1 3591 . . . . . . 7  |-  ( U  i^i  Cat )  C_  U
10 catcoppccl.c . . . . . . . . 9  |-  C  =  (CatCat `  U )
11 catcoppccl.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11, 8catcbas 14986 . . . . . . . 8  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
131, 12eleqtrd 2519 . . . . . . 7  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
149, 13sseldi 3375 . . . . . 6  |-  ( ph  ->  X  e.  U )
15 df-hom 14283 . . . . . . . 8  |-  Hom  = Slot ; 1 4
16 catcoppccl.2 . . . . . . . . 9  |-  ( ph  ->  om  e.  U )
178, 16wunndx 14211 . . . . . . . 8  |-  ( ph  ->  ndx  e.  U )
1815, 8, 17wunstr 14214 . . . . . . 7  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
1915, 8, 14wunstr 14214 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  X
)  e.  U )
208, 19wuntpos 8922 . . . . . . 7  |-  ( ph  -> tpos  ( Hom  `  X
)  e.  U )
218, 18, 20wunop 8910 . . . . . 6  |-  ( ph  -> 
<. ( Hom  `  ndx ) , tpos  ( Hom  `  X ) >.  e.  U
)
228, 14, 21wunsets 14222 . . . . 5  |-  ( ph  ->  ( X sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  X
) >. )  e.  U
)
23 df-cco 14284 . . . . . . 7  |- comp  = Slot ; 1 5
2423, 8, 17wunstr 14214 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
25 df-base 14200 . . . . . . . . . 10  |-  Base  = Slot  1
2625, 8, 14wunstr 14214 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
278, 26, 26wunxp 8912 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  X ) )  e.  U )
288, 27, 26wunxp 8912 . . . . . . 7  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  X ) )  X.  ( Base `  X
) )  e.  U
)
2923, 8, 14wunstr 14214 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
308, 29wunrn 8917 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
318, 30wununi 8894 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
328, 31wundm 8916 . . . . . . . . . . 11  |-  ( ph  ->  dom  U. ran  (comp `  X )  e.  U
)
338, 32wuncnv 8918 . . . . . . . . . 10  |-  ( ph  ->  `' dom  U. ran  (comp `  X )  e.  U
)
348wun0 8906 . . . . . . . . . . 11  |-  ( ph  -> 
(/)  e.  U )
358, 34wunsn 8904 . . . . . . . . . 10  |-  ( ph  ->  { (/) }  e.  U
)
368, 33, 35wunun 8898 . . . . . . . . 9  |-  ( ph  ->  ( `' dom  U. ran  (comp `  X )  u.  { (/) } )  e.  U )
378, 31wunrn 8917 . . . . . . . . 9  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
388, 36, 37wunxp 8912 . . . . . . . 8  |-  ( ph  ->  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  e.  U )
398, 38wunpw 8895 . . . . . . 7  |-  ( ph  ->  ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
)  e.  U )
40 tposssxp 6770 . . . . . . . . . . . 12  |- tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ( ( `' dom  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) )  u.  { (/)
} )  X.  ran  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) ) )
41 ovssunirn 6138 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )
42 dmss 5060 . . . . . . . . . . . . . . 15  |-  ( (
<. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )  ->  dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  C_  dom  U.
ran  (comp `  X )
)
4341, 42ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  ( <. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  dom  U. ran  (comp `  X )
44 cnvss 5033 . . . . . . . . . . . . . 14  |-  ( dom  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) )  C_  dom  U.
ran  (comp `  X )  ->  `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  C_  `' dom  U. ran  (comp `  X ) )
45 unss1 3546 . . . . . . . . . . . . . 14  |-  ( `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  C_  `' dom  U. ran  (comp `  X )  ->  ( `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  u.  { (/)
} )  C_  ( `' dom  U. ran  (comp `  X )  u.  { (/)
} ) )
4643, 44, 45mp2b 10 . . . . . . . . . . . . 13  |-  ( `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  u.  { (/)
} )  C_  ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )
47 rnss 5089 . . . . . . . . . . . . . 14  |-  ( (
<. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )  ->  ran  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  C_  ran  U.
ran  (comp `  X )
)
4841, 47ax-mp 5 . . . . . . . . . . . . 13  |-  ran  ( <. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ran  U. ran  (comp `  X )
49 xpss12 4966 . . . . . . . . . . . . 13  |-  ( ( ( `' dom  ( <. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  u. 
{ (/) } )  C_  ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  /\  ran  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) )  C_  ran  U.
ran  (comp `  X )
)  ->  ( ( `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  u.  { (/)
} )  X.  ran  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) ) )
5046, 48, 49mp2an 672 . . . . . . . . . . . 12  |-  ( ( `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  u.  { (/)
} )  X.  ran  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )
5140, 50sstri 3386 . . . . . . . . . . 11  |- tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )
52 elpw2g 4476 . . . . . . . . . . . 12  |-  ( ( ( `' dom  U. ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  e.  U  ->  (tpos  ( <. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  e. 
~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
)  <-> tpos  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) )  C_  (
( `' dom  U. ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) ) ) )
5338, 52syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (tpos  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  e.  ~P ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  <-> tpos  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  C_  (
( `' dom  U. ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) ) ) )
5451, 53mpbiri 233 . . . . . . . . . 10  |-  ( ph  -> tpos  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) )  e.  ~P ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) ) )
5554ralrimivw 2821 . . . . . . . . 9  |-  ( ph  ->  A. y  e.  (
Base `  X )tpos  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) )  e.  ~P ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) ) )
5655ralrimivw 2821 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) A. y  e.  ( Base `  X )tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  e. 
~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
) )
57 eqid 2443 . . . . . . . . 9  |-  ( x  e.  ( ( Base `  X )  X.  ( Base `  X ) ) ,  y  e.  (
Base `  X )  |-> tpos  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) ) )  =  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) )
5857fmpt2 6662 . . . . . . . 8  |-  ( A. x  e.  ( ( Base `  X )  X.  ( Base `  X
) ) A. y  e.  ( Base `  X
)tpos  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  e.  ~P ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  <->  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) ) : ( ( (
Base `  X )  X.  ( Base `  X
) )  X.  ( Base `  X ) ) --> ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
) )
5956, 58sylib 196 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) ) : ( ( (
Base `  X )  X.  ( Base `  X
) )  X.  ( Base `  X ) ) --> ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
) )
608, 28, 39, 59wunf 8915 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) )  e.  U )
618, 24, 60wunop 8910 . . . . 5  |-  ( ph  -> 
<. (comp `  ndx ) ,  ( x  e.  ( ( Base `  X
)  X.  ( Base `  X ) ) ,  y  e.  ( Base `  X )  |-> tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) ) )
>.  e.  U )
628, 22, 61wunsets 14222 . . . 4  |-  ( ph  ->  ( ( X sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  X
) >. ) sSet  <. (comp ` 
ndx ) ,  ( x  e.  ( (
Base `  X )  X.  ( Base `  X
) ) ,  y  e.  ( Base `  X
)  |-> tpos  ( <. y ,  ( 2nd `  x )
>. (comp `  X )
( 1st `  x
) ) ) >.
)  e.  U )
637, 62eqeltrd 2517 . . 3  |-  ( ph  ->  O  e.  U )
64 inss2 3592 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
6564, 13sseldi 3375 . . . 4  |-  ( ph  ->  X  e.  Cat )
665oppccat 14682 . . . 4  |-  ( X  e.  Cat  ->  O  e.  Cat )
6765, 66syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
6863, 67elind 3561 . 2  |-  ( ph  ->  O  e.  ( U  i^i  Cat ) )
6968, 12eleqtrrd 2520 1  |-  ( ph  ->  O  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2736    u. cun 3347    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   <.cop 3904   U.cuni 4112    X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   omcom 6497   1stc1st 6596   2ndc2nd 6597  tpos ctpos 6765  WUnicwun 8888   1c1 9304   4c4 10394   5c5 10395  ;cdc 10776   ndxcnx 14192   sSet csts 14193   Basecbs 14195   Hom chom 14270  compcco 14271   Catccat 14623  oppCatcoppc 14671  CatCatccatc 14983
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ec 7124  df-qs 7128  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-wun 8890  df-ni 9062  df-pli 9063  df-mi 9064  df-lti 9065  df-plpq 9098  df-mpq 9099  df-ltpq 9100  df-enq 9101  df-nq 9102  df-erq 9103  df-plq 9104  df-mq 9105  df-1nq 9106  df-rq 9107  df-ltnq 9108  df-np 9171  df-plp 9173  df-ltp 9175  df-enr 9247  df-nr 9248  df-c 9309  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-hom 14283  df-cco 14284  df-cat 14627  df-cid 14628  df-oppc 14672  df-catc 14984
This theorem is referenced by: (None)
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