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Theorem catcoppccl 13956
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 2296 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2296 . . . . . 6  |-  (  Hom  `  X )  =  (  Hom  `  X )
4 eqid 2296 . . . . . 6  |-  (comp `  X )  =  (comp `  X )
5 catcoppccl.o . . . . . 6  |-  O  =  (oppCat `  X )
62, 3, 4, 5oppcval 13632 . . . . 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 15 . . . 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 3402 . . . . . . 7  |-  ( U  i^i  Cat )  C_  U
10 catcoppccl.c . . . . . . . . 9  |-  C  =  (CatCat `  U )
11 catcoppccl.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11, 8catcbas 13945 . . . . . . . 8  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
131, 12eleqtrd 2372 . . . . . . 7  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
149, 13sseldi 3191 . . . . . 6  |-  ( ph  ->  X  e.  U )
15 df-hom 13248 . . . . . . . 8  |-  Hom  = Slot ; 1 4
16 catcoppccl.2 . . . . . . . . 9  |-  ( ph  ->  om  e.  U )
178, 16wunndx 13180 . . . . . . . 8  |-  ( ph  ->  ndx  e.  U )
1815, 8, 17wunstr 13183 . . . . . . 7  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
1915, 8, 14wunstr 13183 . . . . . . . 8  |-  ( ph  ->  (  Hom  `  X
)  e.  U )
208, 19wuntpos 8372 . . . . . . 7  |-  ( ph  -> tpos  (  Hom  `  X
)  e.  U )
218, 18, 20wunop 8360 . . . . . 6  |-  ( ph  -> 
<. (  Hom  `  ndx ) , tpos  (  Hom  `  X
) >.  e.  U )
228, 14, 21wunsets 13189 . . . . 5  |-  ( ph  ->  ( X sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  X
) >. )  e.  U
)
23 df-cco 13249 . . . . . . 7  |- comp  = Slot ; 1 5
2423, 8, 17wunstr 13183 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
25 df-base 13169 . . . . . . . . . 10  |-  Base  = Slot  1
2625, 8, 14wunstr 13183 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
278, 26, 26wunxp 8362 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  X ) )  e.  U )
288, 27, 26wunxp 8362 . . . . . . 7  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  X ) )  X.  ( Base `  X
) )  e.  U
)
2923, 8, 14wunstr 13183 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
308, 29wunrn 8367 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
318, 30wununi 8344 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
328, 31wundm 8366 . . . . . . . . . . 11  |-  ( ph  ->  dom  U. ran  (comp `  X )  e.  U
)
338, 32wuncnv 8368 . . . . . . . . . 10  |-  ( ph  ->  `' dom  U. ran  (comp `  X )  e.  U
)
348wun0 8356 . . . . . . . . . . 11  |-  ( ph  -> 
(/)  e.  U )
358, 34wunsn 8354 . . . . . . . . . 10  |-  ( ph  ->  { (/) }  e.  U
)
368, 33, 35wunun 8348 . . . . . . . . 9  |-  ( ph  ->  ( `' dom  U. ran  (comp `  X )  u.  { (/) } )  e.  U )
378, 31wunrn 8367 . . . . . . . . 9  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
388, 36, 37wunxp 8362 . . . . . . . 8  |-  ( ph  ->  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  e.  U )
398, 38wunpw 8345 . . . . . . 7  |-  ( ph  ->  ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
)  e.  U )
40 tposssxp 6254 . . . . . . . . . . . 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 5900 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )
42 dmss 4894 . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . 14  |-  dom  ( <. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  dom  U. ran  (comp `  X )
44 cnvss 4870 . . . . . . . . . . . . . 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 3357 . . . . . . . . . . . . . 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 9 . . . . . . . . . . . . 13  |-  ( `' dom  ( <. y ,  ( 2nd `  x
) >. (comp `  X
) ( 1st `  x
) )  u.  { (/)
} )  C_  ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )
47 rnss 4923 . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . 13  |-  ran  ( <. y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ran  U. ran  (comp `  X )
49 xpss12 4808 . . . . . . . . . . . . 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 653 . . . . . . . . . . . 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 3201 . . . . . . . . . . 11  |- tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )
52 elpw2g 4190 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 224 . . . . . . . . . 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 2640 . . . . . . . . 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 2640 . . . . . . . 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 2296 . . . . . . . . 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 6207 . . . . . . . 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 188 . . . . . . 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 8365 . . . . . 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 8360 . . . . 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 13189 . . . 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 2370 . . 3  |-  ( ph  ->  O  e.  U )
64 inss2 3403 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
6564, 13sseldi 3191 . . . 4  |-  ( ph  ->  X  e.  Cat )
665oppccat 13641 . . . 4  |-  ( X  e.  Cat  ->  O  e.  Cat )
6765, 66syl 15 . . 3  |-  ( ph  ->  O  e.  Cat )
68 elin 3371 . . 3  |-  ( O  e.  ( U  i^i  Cat )  <->  ( O  e.  U  /\  O  e. 
Cat ) )
6963, 67, 68sylanbrc 645 . 2  |-  ( ph  ->  O  e.  ( U  i^i  Cat ) )
7069, 12eleqtrrd 2373 1  |-  ( ph  ->  O  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   <.cop 3656   U.cuni 3843   omcom 4672    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137  tpos ctpos 6249  WUnicwun 8338   1c1 8754   4c4 9813   5c5 9814  ;cdc 10140   ndxcnx 13161   sSet csts 13162   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582  oppCatcoppc 13630  CatCatccatc 13942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-wun 8340  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-ltp 8625  df-enr 8697  df-nr 8698  df-c 8759  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-oppc 13631  df-catc 13943
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