Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  issubcata Unicode version

Theorem issubcata 25846
Description: The predicate "is a subcategory of"  T. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
issubcata.2  |-  D  =  ( dom_ `  T
)
issubcata.3  |-  C  =  ( cod_ `  T
)
issubcata.4  |-  R  =  ( o_ `  T
)
issubcata.5  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
issubcata  |-  ( T  e.  Cat OLD  ->  ( U  e.  (  SubCat  `  T )  <->  ( U  e.  Cat OLD  /\  (
( id_ `  U
)  C_  J  /\  ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
o_ `  U )  C_  R ) ) ) )

Proof of Theorem issubcata
StepHypRef Expression
1 issubcata.2 . . . 4  |-  D  =  ( dom_ `  T
)
2 issubcata.3 . . . 4  |-  C  =  ( cod_ `  T
)
3 issubcata.4 . . . 4  |-  R  =  ( o_ `  T
)
4 issubcata.5 . . . 4  |-  J  =  ( id_ `  T
)
51, 2, 3, 4issubcat 25845 . . 3  |-  ( T  e.  Cat OLD  ->  ( 
SubCat  `  T )  =  (  Cat OLD  i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) ) )
65eleq2d 2350 . 2  |-  ( T  e.  Cat OLD  ->  ( U  e.  (  SubCat  `  T )  <->  U  e.  (  Cat OLD  i^i  (
( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) ) ) ) )
7 elin 3358 . . 3  |-  ( U  e.  (  Cat OLD  i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) )  <-> 
( U  e.  Cat OLD 
/\  U  e.  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) ) ) )
8 strcat 25760 . . . . . . . 8  |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
98sseli 3176 . . . . . . 7  |-  ( U  e.  Cat OLD  ->  U  e.  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) ) )
10 1st2nd2 6159 . . . . . . 7  |-  ( U  e.  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
11 elxp6 6151 . . . . . . . 8  |-  ( U  e.  ( ( ~P D  X.  ~P C
)  X.  ( ~P J  X.  ~P R
) )  <->  ( U  =  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  /\  (
( 1st `  U
)  e.  ( ~P D  X.  ~P C
)  /\  ( 2nd `  U )  e.  ( ~P J  X.  ~P R ) ) ) )
1211baib 871 . . . . . . 7  |-  ( U  =  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  ->  ( U  e.  ( ( ~P D  X.  ~P C
)  X.  ( ~P J  X.  ~P R
) )  <->  ( ( 1st `  U )  e.  ( ~P D  X.  ~P C )  /\  ( 2nd `  U )  e.  ( ~P J  X.  ~P R ) ) ) )
139, 10, 123syl 18 . . . . . 6  |-  ( U  e.  Cat OLD  ->  ( U  e.  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) )  <->  ( ( 1st `  U )  e.  ( ~P D  X.  ~P C )  /\  ( 2nd `  U )  e.  ( ~P J  X.  ~P R ) ) ) )
14 xp1st 6149 . . . . . . . . . 10  |-  ( U  e.  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )  ->  ( 1st `  U
)  e.  ( _V 
X.  _V ) )
159, 14syl 15 . . . . . . . . 9  |-  ( U  e.  Cat OLD  ->  ( 1st `  U )  e.  ( _V  X.  _V ) )
16 1st2nd2 6159 . . . . . . . . 9  |-  ( ( 1st `  U )  e.  ( _V  X.  _V )  ->  ( 1st `  U )  =  <. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
17 elxp6 6151 . . . . . . . . . 10  |-  ( ( 1st `  U )  e.  ( ~P D  X.  ~P C )  <->  ( ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>.  /\  ( ( 1st `  ( 1st `  U
) )  e.  ~P D  /\  ( 2nd `  ( 1st `  U ) )  e.  ~P C ) ) )
1817baib 871 . . . . . . . . 9  |-  ( ( 1st `  U )  =  <. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>.  ->  ( ( 1st `  U )  e.  ( ~P D  X.  ~P C )  <->  ( ( 1st `  ( 1st `  U
) )  e.  ~P D  /\  ( 2nd `  ( 1st `  U ) )  e.  ~P C ) ) )
1915, 16, 183syl 18 . . . . . . . 8  |-  ( U  e.  Cat OLD  ->  ( ( 1st `  U
)  e.  ( ~P D  X.  ~P C
)  <->  ( ( 1st `  ( 1st `  U
) )  e.  ~P D  /\  ( 2nd `  ( 1st `  U ) )  e.  ~P C ) ) )
20 eqid 2283 . . . . . . . . . . . 12  |-  ( dom_ `  U )  =  (
dom_ `  U )
2120domval 25723 . . . . . . . . . . 11  |-  ( dom_ `  U )  =  ( 1st `  ( 1st `  U ) )
2221eleq1i 2346 . . . . . . . . . 10  |-  ( (
dom_ `  U )  e.  ~P D  <->  ( 1st `  ( 1st `  U
) )  e.  ~P D )
23 fvex 5539 . . . . . . . . . . 11  |-  ( dom_ `  U )  e.  _V
2423elpw 3631 . . . . . . . . . 10  |-  ( (
dom_ `  U )  e.  ~P D  <->  ( dom_ `  U )  C_  D
)
2522, 24bitr3i 242 . . . . . . . . 9  |-  ( ( 1st `  ( 1st `  U ) )  e. 
~P D  <->  ( dom_ `  U )  C_  D
)
26 eqid 2283 . . . . . . . . . . . 12  |-  ( cod_ `  U )  =  (
cod_ `  U )
2726codval 25724 . . . . . . . . . . 11  |-  ( cod_ `  U )  =  ( 2nd `  ( 1st `  U ) )
2827eleq1i 2346 . . . . . . . . . 10  |-  ( (
cod_ `  U )  e.  ~P C  <->  ( 2nd `  ( 1st `  U
) )  e.  ~P C )
29 fvex 5539 . . . . . . . . . . 11  |-  ( cod_ `  U )  e.  _V
3029elpw 3631 . . . . . . . . . 10  |-  ( (
cod_ `  U )  e.  ~P C  <->  ( cod_ `  U )  C_  C
)
3128, 30bitr3i 242 . . . . . . . . 9  |-  ( ( 2nd `  ( 1st `  U ) )  e. 
~P C  <->  ( cod_ `  U )  C_  C
)
3225, 31anbi12i 678 . . . . . . . 8  |-  ( ( ( 1st `  ( 1st `  U ) )  e.  ~P D  /\  ( 2nd `  ( 1st `  U ) )  e. 
~P C )  <->  ( ( dom_ `  U )  C_  D  /\  ( cod_ `  U
)  C_  C )
)
3319, 32syl6bb 252 . . . . . . 7  |-  ( U  e.  Cat OLD  ->  ( ( 1st `  U
)  e.  ( ~P D  X.  ~P C
)  <->  ( ( dom_ `  U )  C_  D  /\  ( cod_ `  U
)  C_  C )
) )
34 xp2nd 6150 . . . . . . . . . 10  |-  ( U  e.  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )  ->  ( 2nd `  U
)  e.  ( _V 
X.  _V ) )
359, 34syl 15 . . . . . . . . 9  |-  ( U  e.  Cat OLD  ->  ( 2nd `  U )  e.  ( _V  X.  _V ) )
36 1st2nd2 6159 . . . . . . . . 9  |-  ( ( 2nd `  U )  e.  ( _V  X.  _V )  ->  ( 2nd `  U )  =  <. ( 1st `  ( 2nd `  U ) ) ,  ( 2nd `  ( 2nd `  U ) )
>. )
37 elxp6 6151 . . . . . . . . . 10  |-  ( ( 2nd `  U )  e.  ( ~P J  X.  ~P R )  <->  ( ( 2nd `  U )  = 
<. ( 1st `  ( 2nd `  U ) ) ,  ( 2nd `  ( 2nd `  U ) )
>.  /\  ( ( 1st `  ( 2nd `  U
) )  e.  ~P J  /\  ( 2nd `  ( 2nd `  U ) )  e.  ~P R ) ) )
3837baib 871 . . . . . . . . 9  |-  ( ( 2nd `  U )  =  <. ( 1st `  ( 2nd `  U ) ) ,  ( 2nd `  ( 2nd `  U ) )
>.  ->  ( ( 2nd `  U )  e.  ( ~P J  X.  ~P R )  <->  ( ( 1st `  ( 2nd `  U
) )  e.  ~P J  /\  ( 2nd `  ( 2nd `  U ) )  e.  ~P R ) ) )
3935, 36, 383syl 18 . . . . . . . 8  |-  ( U  e.  Cat OLD  ->  ( ( 2nd `  U
)  e.  ( ~P J  X.  ~P R
)  <->  ( ( 1st `  ( 2nd `  U
) )  e.  ~P J  /\  ( 2nd `  ( 2nd `  U ) )  e.  ~P R ) ) )
40 eqid 2283 . . . . . . . . . . . 12  |-  ( id_ `  U )  =  ( id_ `  U )
4140idval 25725 . . . . . . . . . . 11  |-  ( id_ `  U )  =  ( 1st `  ( 2nd `  U ) )
4241eleq1i 2346 . . . . . . . . . 10  |-  ( ( id_ `  U )  e.  ~P J  <->  ( 1st `  ( 2nd `  U
) )  e.  ~P J )
43 fvex 5539 . . . . . . . . . . 11  |-  ( id_ `  U )  e.  _V
4443elpw 3631 . . . . . . . . . 10  |-  ( ( id_ `  U )  e.  ~P J  <->  ( id_ `  U )  C_  J
)
4542, 44bitr3i 242 . . . . . . . . 9  |-  ( ( 1st `  ( 2nd `  U ) )  e. 
~P J  <->  ( id_ `  U )  C_  J
)
46 eqid 2283 . . . . . . . . . . . 12  |-  ( o_
`  U )  =  ( o_ `  U
)
4746cmpval 25726 . . . . . . . . . . 11  |-  ( o_
`  U )  =  ( 2nd `  ( 2nd `  U ) )
4847eleq1i 2346 . . . . . . . . . 10  |-  ( ( o_ `  U )  e.  ~P R  <->  ( 2nd `  ( 2nd `  U
) )  e.  ~P R )
49 fvex 5539 . . . . . . . . . . 11  |-  ( o_
`  U )  e. 
_V
5049elpw 3631 . . . . . . . . . 10  |-  ( ( o_ `  U )  e.  ~P R  <->  ( o_ `  U )  C_  R
)
5148, 50bitr3i 242 . . . . . . . . 9  |-  ( ( 2nd `  ( 2nd `  U ) )  e. 
~P R  <->  ( o_ `  U )  C_  R
)
5245, 51anbi12i 678 . . . . . . . 8  |-  ( ( ( 1st `  ( 2nd `  U ) )  e.  ~P J  /\  ( 2nd `  ( 2nd `  U ) )  e. 
~P R )  <->  ( ( id_ `  U )  C_  J  /\  ( o_ `  U )  C_  R
) )
5339, 52syl6bb 252 . . . . . . 7  |-  ( U  e.  Cat OLD  ->  ( ( 2nd `  U
)  e.  ( ~P J  X.  ~P R
)  <->  ( ( id_ `  U )  C_  J  /\  ( o_ `  U
)  C_  R )
) )
5433, 53anbi12d 691 . . . . . 6  |-  ( U  e.  Cat OLD  ->  ( ( ( 1st `  U
)  e.  ( ~P D  X.  ~P C
)  /\  ( 2nd `  U )  e.  ( ~P J  X.  ~P R ) )  <->  ( (
( dom_ `  U )  C_  D  /\  ( cod_ `  U )  C_  C
)  /\  ( ( id_ `  U )  C_  J  /\  ( o_ `  U )  C_  R
) ) ) )
5513, 54bitrd 244 . . . . 5  |-  ( U  e.  Cat OLD  ->  ( U  e.  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) )  <->  ( (
( dom_ `  U )  C_  D  /\  ( cod_ `  U )  C_  C
)  /\  ( ( id_ `  U )  C_  J  /\  ( o_ `  U )  C_  R
) ) ) )
56 3anass 938 . . . . . 6  |-  ( ( ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  ( id_ `  U )  C_  J  /\  ( o_ `  U )  C_  R
)  <->  ( ( (
dom_ `  U )  C_  D  /\  ( cod_ `  U )  C_  C
)  /\  ( ( id_ `  U )  C_  J  /\  ( o_ `  U )  C_  R
) ) )
57 3ancoma 941 . . . . . 6  |-  ( ( ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  ( id_ `  U )  C_  J  /\  ( o_ `  U )  C_  R
)  <->  ( ( id_ `  U )  C_  J  /\  ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
o_ `  U )  C_  R ) )
5856, 57bitr3i 242 . . . . 5  |-  ( ( ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
( id_ `  U
)  C_  J  /\  ( o_ `  U ) 
C_  R ) )  <-> 
( ( id_ `  U
)  C_  J  /\  ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
o_ `  U )  C_  R ) )
5955, 58syl6bb 252 . . . 4  |-  ( U  e.  Cat OLD  ->  ( U  e.  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) )  <->  ( ( id_ `  U )  C_  J  /\  ( ( dom_ `  U )  C_  D  /\  ( cod_ `  U
)  C_  C )  /\  ( o_ `  U
)  C_  R )
) )
6059pm5.32i 618 . . 3  |-  ( ( U  e.  Cat OLD  /\  U  e.  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) ) )  <->  ( U  e.  Cat OLD  /\  (
( id_ `  U
)  C_  J  /\  ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
o_ `  U )  C_  R ) ) )
617, 60bitri 240 . 2  |-  ( U  e.  (  Cat OLD  i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) )  <-> 
( U  e.  Cat OLD 
/\  ( ( id_ `  U )  C_  J  /\  ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
o_ `  U )  C_  R ) ) )
626, 61syl6bb 252 1  |-  ( T  e.  Cat OLD  ->  ( U  e.  (  SubCat  `  T )  <->  ( U  e.  Cat OLD  /\  (
( id_ `  U
)  C_  J  /\  ( ( dom_ `  U
)  C_  D  /\  ( cod_ `  U )  C_  C )  /\  (
o_ `  U )  C_  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643    X. cxp 4687   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752    SubCat csubcat 25843
This theorem is referenced by:  issubcatb  25847  obsubc2  25850  idsubc  25851  domsubc  25852  codsubc  25853  subctct  25854  morsubc  25855  cmpsubc  25856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-catOLD 25753  df-subcat 25844
  Copyright terms: Public domain W3C validator