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Theorem catcoppccl 16052
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 2462 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2462 . . . . . 6  |-  ( Hom  `  X )  =  ( Hom  `  X )
4 eqid 2462 . . . . . 6  |-  (comp `  X )  =  (comp `  X )
5 catcoppccl.o . . . . . 6  |-  O  =  (oppCat `  X )
62, 3, 4, 5oppcval 15667 . . . . 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 17 . . . 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 3664 . . . . . . 7  |-  ( U  i^i  Cat )  C_  U
10 catcoppccl.c . . . . . . . . 9  |-  C  =  (CatCat `  U )
11 catcoppccl.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11, 8catcbas 16041 . . . . . . . 8  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
131, 12eleqtrd 2542 . . . . . . 7  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
149, 13sseldi 3442 . . . . . 6  |-  ( ph  ->  X  e.  U )
15 df-hom 15263 . . . . . . . 8  |-  Hom  = Slot ; 1 4
16 catcoppccl.2 . . . . . . . . 9  |-  ( ph  ->  om  e.  U )
178, 16wunndx 15186 . . . . . . . 8  |-  ( ph  ->  ndx  e.  U )
1815, 8, 17wunstr 15189 . . . . . . 7  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
1915, 8, 14wunstr 15189 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  X
)  e.  U )
208, 19wuntpos 9185 . . . . . . 7  |-  ( ph  -> tpos  ( Hom  `  X
)  e.  U )
218, 18, 20wunop 9173 . . . . . 6  |-  ( ph  -> 
<. ( Hom  `  ndx ) , tpos  ( Hom  `  X ) >.  e.  U
)
228, 14, 21wunsets 15199 . . . . 5  |-  ( ph  ->  ( X sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  X
) >. )  e.  U
)
23 df-cco 15264 . . . . . . 7  |- comp  = Slot ; 1 5
2423, 8, 17wunstr 15189 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
25 df-base 15175 . . . . . . . . . 10  |-  Base  = Slot  1
2625, 8, 14wunstr 15189 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
278, 26, 26wunxp 9175 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  X ) )  e.  U )
288, 27, 26wunxp 9175 . . . . . . 7  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  X ) )  X.  ( Base `  X
) )  e.  U
)
2923, 8, 14wunstr 15189 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
308, 29wunrn 9180 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
318, 30wununi 9157 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
328, 31wundm 9179 . . . . . . . . . . 11  |-  ( ph  ->  dom  U. ran  (comp `  X )  e.  U
)
338, 32wuncnv 9181 . . . . . . . . . 10  |-  ( ph  ->  `' dom  U. ran  (comp `  X )  e.  U
)
348wun0 9169 . . . . . . . . . . 11  |-  ( ph  -> 
(/)  e.  U )
358, 34wunsn 9167 . . . . . . . . . 10  |-  ( ph  ->  { (/) }  e.  U
)
368, 33, 35wunun 9161 . . . . . . . . 9  |-  ( ph  ->  ( `' dom  U. ran  (comp `  X )  u.  { (/) } )  e.  U )
378, 31wunrn 9180 . . . . . . . . 9  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
388, 36, 37wunxp 9175 . . . . . . . 8  |-  ( ph  ->  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  e.  U )
398, 38wunpw 9158 . . . . . . 7  |-  ( ph  ->  ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
)  e.  U )
40 tposssxp 7003 . . . . . . . . . . . 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 6344 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )
42 dmss 5053 . . . . . . . . . . . . . . 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 5026 . . . . . . . . . . . . . 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 3615 . . . . . . . . . . . . . 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 5082 . . . . . . . . . . . . . 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 4959 . . . . . . . . . . . . 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 683 . . . . . . . . . . . 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 3453 . . . . . . . . . . 11  |- tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )
52 elpw2g 4580 . . . . . . . . . . . 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 17 . . . . . . . . . . 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 241 . . . . . . . . . 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 2815 . . . . . . . . 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 2815 . . . . . . . 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 2462 . . . . . . . . 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 6887 . . . . . . . 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 201 . . . . . . 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 9178 . . . . . 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 9173 . . . . 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 15199 . . . 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 2540 . . 3  |-  ( ph  ->  O  e.  U )
64 inss2 3665 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
6564, 13sseldi 3442 . . . 4  |-  ( ph  ->  X  e.  Cat )
665oppccat 15676 . . . 4  |-  ( X  e.  Cat  ->  O  e.  Cat )
6765, 66syl 17 . . 3  |-  ( ph  ->  O  e.  Cat )
6863, 67elind 3630 . 2  |-  ( ph  ->  O  e.  ( U  i^i  Cat ) )
6968, 12eleqtrrd 2543 1  |-  ( ph  ->  O  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1455    e. wcel 1898   A.wral 2749    u. cun 3414    i^i cin 3415    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   {csn 3980   <.cop 3986   U.cuni 4212    X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854   -->wf 5597   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317   omcom 6719   1stc1st 6818   2ndc2nd 6819  tpos ctpos 6998  WUnicwun 9151   1c1 9566   4c4 10689   5c5 10690  ;cdc 11080   ndxcnx 15167   sSet csts 15168   Basecbs 15170   Hom chom 15250  compcco 15251   Catccat 15619  oppCatcoppc 15665  CatCatccatc 16038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-tpos 6999  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-omul 7213  df-er 7389  df-ec 7391  df-qs 7395  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-wun 9153  df-ni 9323  df-pli 9324  df-mi 9325  df-lti 9326  df-plpq 9359  df-mpq 9360  df-ltpq 9361  df-enq 9362  df-nq 9363  df-erq 9364  df-plq 9365  df-mq 9366  df-1nq 9367  df-rq 9368  df-ltnq 9369  df-np 9432  df-plp 9434  df-ltp 9436  df-enr 9506  df-nr 9507  df-c 9571  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-fz 11814  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-hom 15263  df-cco 15264  df-cat 15623  df-cid 15624  df-oppc 15666  df-catc 16039
This theorem is referenced by: (None)
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