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Theorem issubcata 25949
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 25948 . . 3  |-  ( T  e.  Cat OLD  ->  ( 
SubCat  `  T )  =  (  Cat OLD  i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) ) )
65eleq2d 2363 . 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 3371 . . 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 25863 . . . . . . . 8  |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
98sseli 3189 . . . . . . 7  |-  ( U  e.  Cat OLD  ->  U  e.  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) ) )
10 1st2nd2 6175 . . . . . . 7  |-  ( U  e.  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
11 elxp6 6167 . . . . . . . 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 6165 . . . . . . . . . 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 6175 . . . . . . . . 9  |-  ( ( 1st `  U )  e.  ( _V  X.  _V )  ->  ( 1st `  U )  =  <. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
17 elxp6 6167 . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( dom_ `  U )  =  (
dom_ `  U )
2120domval 25826 . . . . . . . . . . 11  |-  ( dom_ `  U )  =  ( 1st `  ( 1st `  U ) )
2221eleq1i 2359 . . . . . . . . . 10  |-  ( (
dom_ `  U )  e.  ~P D  <->  ( 1st `  ( 1st `  U
) )  e.  ~P D )
23 fvex 5555 . . . . . . . . . . 11  |-  ( dom_ `  U )  e.  _V
2423elpw 3644 . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( cod_ `  U )  =  (
cod_ `  U )
2726codval 25827 . . . . . . . . . . 11  |-  ( cod_ `  U )  =  ( 2nd `  ( 1st `  U ) )
2827eleq1i 2359 . . . . . . . . . 10  |-  ( (
cod_ `  U )  e.  ~P C  <->  ( 2nd `  ( 1st `  U
) )  e.  ~P C )
29 fvex 5555 . . . . . . . . . . 11  |-  ( cod_ `  U )  e.  _V
3029elpw 3644 . . . . . . . . . 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 6166 . . . . . . . . . 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 6175 . . . . . . . . 9  |-  ( ( 2nd `  U )  e.  ( _V  X.  _V )  ->  ( 2nd `  U )  =  <. ( 1st `  ( 2nd `  U ) ) ,  ( 2nd `  ( 2nd `  U ) )
>. )
37 elxp6 6167 . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( id_ `  U )  =  ( id_ `  U )
4140idval 25828 . . . . . . . . . . 11  |-  ( id_ `  U )  =  ( 1st `  ( 2nd `  U ) )
4241eleq1i 2359 . . . . . . . . . 10  |-  ( ( id_ `  U )  e.  ~P J  <->  ( 1st `  ( 2nd `  U
) )  e.  ~P J )
43 fvex 5555 . . . . . . . . . . 11  |-  ( id_ `  U )  e.  _V
4443elpw 3644 . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( o_
`  U )  =  ( o_ `  U
)
4746cmpval 25829 . . . . . . . . . . 11  |-  ( o_
`  U )  =  ( 2nd `  ( 2nd `  U ) )
4847eleq1i 2359 . . . . . . . . . 10  |-  ( ( o_ `  U )  e.  ~P R  <->  ( 2nd `  ( 2nd `  U
) )  e.  ~P R )
49 fvex 5555 . . . . . . . . . . 11  |-  ( o_
`  U )  e. 
_V
5049elpw 3644 . . . . . . . . . 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   <.cop 3656    X. cxp 4703   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818    Cat
OLD ccatOLD 25855    SubCat csubcat 25946
This theorem is referenced by:  issubcatb  25950  obsubc2  25953  idsubc  25954  domsubc  25955  codsubc  25956  subctct  25957  morsubc  25958  cmpsubc  25959
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-catOLD 25856  df-subcat 25947
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