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Theorem catcoppccl 15309
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 2467 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2467 . . . . . 6  |-  ( Hom  `  X )  =  ( Hom  `  X )
4 eqid 2467 . . . . . 6  |-  (comp `  X )  =  (comp `  X )
5 catcoppccl.o . . . . . 6  |-  O  =  (oppCat `  X )
62, 3, 4, 5oppcval 14985 . . . . 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 3723 . . . . . . 7  |-  ( U  i^i  Cat )  C_  U
10 catcoppccl.c . . . . . . . . 9  |-  C  =  (CatCat `  U )
11 catcoppccl.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11, 8catcbas 15298 . . . . . . . 8  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
131, 12eleqtrd 2557 . . . . . . 7  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
149, 13sseldi 3507 . . . . . 6  |-  ( ph  ->  X  e.  U )
15 df-hom 14595 . . . . . . . 8  |-  Hom  = Slot ; 1 4
16 catcoppccl.2 . . . . . . . . 9  |-  ( ph  ->  om  e.  U )
178, 16wunndx 14522 . . . . . . . 8  |-  ( ph  ->  ndx  e.  U )
1815, 8, 17wunstr 14525 . . . . . . 7  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
1915, 8, 14wunstr 14525 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  X
)  e.  U )
208, 19wuntpos 9124 . . . . . . 7  |-  ( ph  -> tpos  ( Hom  `  X
)  e.  U )
218, 18, 20wunop 9112 . . . . . 6  |-  ( ph  -> 
<. ( Hom  `  ndx ) , tpos  ( Hom  `  X ) >.  e.  U
)
228, 14, 21wunsets 14533 . . . . 5  |-  ( ph  ->  ( X sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  X
) >. )  e.  U
)
23 df-cco 14596 . . . . . . 7  |- comp  = Slot ; 1 5
2423, 8, 17wunstr 14525 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
25 df-base 14511 . . . . . . . . . 10  |-  Base  = Slot  1
2625, 8, 14wunstr 14525 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
278, 26, 26wunxp 9114 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  X ) )  e.  U )
288, 27, 26wunxp 9114 . . . . . . 7  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  X ) )  X.  ( Base `  X
) )  e.  U
)
2923, 8, 14wunstr 14525 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
308, 29wunrn 9119 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
318, 30wununi 9096 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
328, 31wundm 9118 . . . . . . . . . . 11  |-  ( ph  ->  dom  U. ran  (comp `  X )  e.  U
)
338, 32wuncnv 9120 . . . . . . . . . 10  |-  ( ph  ->  `' dom  U. ran  (comp `  X )  e.  U
)
348wun0 9108 . . . . . . . . . . 11  |-  ( ph  -> 
(/)  e.  U )
358, 34wunsn 9106 . . . . . . . . . 10  |-  ( ph  ->  { (/) }  e.  U
)
368, 33, 35wunun 9100 . . . . . . . . 9  |-  ( ph  ->  ( `' dom  U. ran  (comp `  X )  u.  { (/) } )  e.  U )
378, 31wunrn 9119 . . . . . . . . 9  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
388, 36, 37wunxp 9114 . . . . . . . 8  |-  ( ph  ->  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  e.  U )
398, 38wunpw 9097 . . . . . . 7  |-  ( ph  ->  ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
)  e.  U )
40 tposssxp 6971 . . . . . . . . . . . 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 6321 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )
42 dmss 5208 . . . . . . . . . . . . . . 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 5181 . . . . . . . . . . . . . 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 3678 . . . . . . . . . . . . . 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 5237 . . . . . . . . . . . . . 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 5114 . . . . . . . . . . . . 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 3518 . . . . . . . . . . 11  |- tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )
52 elpw2g 4616 . . . . . . . . . . . 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 2882 . . . . . . . . 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 2882 . . . . . . . 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 2467 . . . . . . . . 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 6862 . . . . . . . 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 9117 . . . . . 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 9112 . . . . 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 14533 . . . 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 2555 . . 3  |-  ( ph  ->  O  e.  U )
64 inss2 3724 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
6564, 13sseldi 3507 . . . 4  |-  ( ph  ->  X  e.  Cat )
665oppccat 14994 . . . 4  |-  ( X  e.  Cat  ->  O  e.  Cat )
6765, 66syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
6863, 67elind 3693 . 2  |-  ( ph  ->  O  e.  ( U  i^i  Cat ) )
6968, 12eleqtrrd 2558 1  |-  ( ph  ->  O  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2817    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   <.cop 4039   U.cuni 4251    X. cxp 5003   `'ccnv 5004   dom cdm 5005   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   omcom 6695   1stc1st 6793   2ndc2nd 6794  tpos ctpos 6966  WUnicwun 9090   1c1 9505   4c4 10599   5c5 10600  ;cdc 10988   ndxcnx 14503   sSet csts 14504   Basecbs 14506   Hom chom 14582  compcco 14583   Catccat 14935  oppCatcoppc 14983  CatCatccatc 15295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-wun 9092  df-ni 9262  df-pli 9263  df-mi 9264  df-lti 9265  df-plpq 9298  df-mpq 9299  df-ltpq 9300  df-enq 9301  df-nq 9302  df-erq 9303  df-plq 9304  df-mq 9305  df-1nq 9306  df-rq 9307  df-ltnq 9308  df-np 9371  df-plp 9373  df-ltp 9375  df-enr 9445  df-nr 9446  df-c 9510  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-hom 14595  df-cco 14596  df-cat 14939  df-cid 14940  df-oppc 14984  df-catc 15296
This theorem is referenced by: (None)
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