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Theorem catcoppccl 14263
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 2436 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2436 . . . . . 6  |-  (  Hom  `  X )  =  (  Hom  `  X )
4 eqid 2436 . . . . . 6  |-  (comp `  X )  =  (comp `  X )
5 catcoppccl.o . . . . . 6  |-  O  =  (oppCat `  X )
62, 3, 4, 5oppcval 13939 . . . . 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 3561 . . . . . . 7  |-  ( U  i^i  Cat )  C_  U
10 catcoppccl.c . . . . . . . . 9  |-  C  =  (CatCat `  U )
11 catcoppccl.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11, 8catcbas 14252 . . . . . . . 8  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
131, 12eleqtrd 2512 . . . . . . 7  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
149, 13sseldi 3346 . . . . . 6  |-  ( ph  ->  X  e.  U )
15 df-hom 13553 . . . . . . . 8  |-  Hom  = Slot ; 1 4
16 catcoppccl.2 . . . . . . . . 9  |-  ( ph  ->  om  e.  U )
178, 16wunndx 13485 . . . . . . . 8  |-  ( ph  ->  ndx  e.  U )
1815, 8, 17wunstr 13488 . . . . . . 7  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
1915, 8, 14wunstr 13488 . . . . . . . 8  |-  ( ph  ->  (  Hom  `  X
)  e.  U )
208, 19wuntpos 8609 . . . . . . 7  |-  ( ph  -> tpos  (  Hom  `  X
)  e.  U )
218, 18, 20wunop 8597 . . . . . 6  |-  ( ph  -> 
<. (  Hom  `  ndx ) , tpos  (  Hom  `  X
) >.  e.  U )
228, 14, 21wunsets 13494 . . . . 5  |-  ( ph  ->  ( X sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  X
) >. )  e.  U
)
23 df-cco 13554 . . . . . . 7  |- comp  = Slot ; 1 5
2423, 8, 17wunstr 13488 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
25 df-base 13474 . . . . . . . . . 10  |-  Base  = Slot  1
2625, 8, 14wunstr 13488 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
278, 26, 26wunxp 8599 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  X ) )  e.  U )
288, 27, 26wunxp 8599 . . . . . . 7  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  X ) )  X.  ( Base `  X
) )  e.  U
)
2923, 8, 14wunstr 13488 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
308, 29wunrn 8604 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
318, 30wununi 8581 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
328, 31wundm 8603 . . . . . . . . . . 11  |-  ( ph  ->  dom  U. ran  (comp `  X )  e.  U
)
338, 32wuncnv 8605 . . . . . . . . . 10  |-  ( ph  ->  `' dom  U. ran  (comp `  X )  e.  U
)
348wun0 8593 . . . . . . . . . . 11  |-  ( ph  -> 
(/)  e.  U )
358, 34wunsn 8591 . . . . . . . . . 10  |-  ( ph  ->  { (/) }  e.  U
)
368, 33, 35wunun 8585 . . . . . . . . 9  |-  ( ph  ->  ( `' dom  U. ran  (comp `  X )  u.  { (/) } )  e.  U )
378, 31wunrn 8604 . . . . . . . . 9  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
388, 36, 37wunxp 8599 . . . . . . . 8  |-  ( ph  ->  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )  e.  U )
398, 38wunpw 8582 . . . . . . 7  |-  ( ph  ->  ~P ( ( `' dom  U. ran  (comp `  X )  u.  { (/)
} )  X.  ran  U.
ran  (comp `  X )
)  e.  U )
40 tposssxp 6483 . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  U.
ran  (comp `  X )
42 dmss 5069 . . . . . . . . . . . . . . 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 5045 . . . . . . . . . . . . . 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 3516 . . . . . . . . . . . . . 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 5098 . . . . . . . . . . . . . 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 4981 . . . . . . . . . . . . 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 654 . . . . . . . . . . . 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 3357 . . . . . . . . . . 11  |- tpos  ( <.
y ,  ( 2nd `  x ) >. (comp `  X ) ( 1st `  x ) )  C_  ( ( `' dom  U.
ran  (comp `  X )  u.  { (/) } )  X. 
ran  U. ran  (comp `  X ) )
52 elpw2g 4363 . . . . . . . . . . . 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 225 . . . . . . . . . 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 2790 . . . . . . . . 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 2790 . . . . . . . 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 2436 . . . . . . . . 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 6418 . . . . . . . 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 189 . . . . . . 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 8602 . . . . . 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 8597 . . . . 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 13494 . . . 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 2510 . . 3  |-  ( ph  ->  O  e.  U )
64 inss2 3562 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
6564, 13sseldi 3346 . . . 4  |-  ( ph  ->  X  e.  Cat )
665oppccat 13948 . . . 4  |-  ( X  e.  Cat  ->  O  e.  Cat )
6765, 66syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
68 elin 3530 . . 3  |-  ( O  e.  ( U  i^i  Cat )  <->  ( O  e.  U  /\  O  e. 
Cat ) )
6963, 67, 68sylanbrc 646 . 2  |-  ( ph  ->  O  e.  ( U  i^i  Cat ) )
7069, 12eleqtrrd 2513 1  |-  ( ph  ->  O  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2705    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   <.cop 3817   U.cuni 4015   omcom 4845    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348  tpos ctpos 6478  WUnicwun 8575   1c1 8991   4c4 10051   5c5 10052  ;cdc 10382   ndxcnx 13466   sSet csts 13467   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889  oppCatcoppc 13937  CatCatccatc 14249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-wun 8577  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-plp 8860  df-ltp 8862  df-enr 8934  df-nr 8935  df-c 8996  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-hom 13553  df-cco 13554  df-cat 13893  df-cid 13894  df-oppc 13938  df-catc 14250
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