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Theorem brssc 13693
Description: The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
brssc  |-  ( H 
C_cat  J  <->  E. t ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) ) )
Distinct variable groups:    t, s, x, H    J, s, t, x

Proof of Theorem brssc
Dummy variables  h  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sscrel 13692 . . 3  |-  Rel  C_cat
2 brrelex12 4728 . . 3  |-  ( ( Rel  C_cat  /\  H  C_cat  J )  ->  ( H  e. 
_V  /\  J  e.  _V ) )
31, 2mpan 651 . 2  |-  ( H 
C_cat  J  ->  ( H  e.  _V  /\  J  e. 
_V ) )
4 vex 2793 . . . . . 6  |-  t  e. 
_V
54, 4xpex 4803 . . . . 5  |-  ( t  X.  t )  e. 
_V
6 fnex 5743 . . . . 5  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
75, 6mpan2 652 . . . 4  |-  ( J  Fn  ( t  X.  t )  ->  J  e.  _V )
8 elex 2798 . . . . 5  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  e.  _V )
98rexlimivw 2665 . . . 4  |-  ( E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  ->  H  e.  _V )
107, 9anim12ci 550 . . 3  |-  ( ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) )  ->  ( H  e. 
_V  /\  J  e.  _V ) )
1110exlimiv 1668 . 2  |-  ( E. t ( J  Fn  ( t  X.  t
)  /\  E. s  e.  ~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) )  -> 
( H  e.  _V  /\  J  e.  _V )
)
12 simpr 447 . . . . . 6  |-  ( ( h  =  H  /\  j  =  J )  ->  j  =  J )
1312fneq1d 5337 . . . . 5  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j  Fn  (
t  X.  t )  <-> 
J  Fn  ( t  X.  t ) ) )
14 simpl 443 . . . . . . 7  |-  ( ( h  =  H  /\  j  =  J )  ->  h  =  H )
1512fveq1d 5529 . . . . . . . . 9  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j `  x
)  =  ( J `
 x ) )
1615pweqd 3632 . . . . . . . 8  |-  ( ( h  =  H  /\  j  =  J )  ->  ~P ( j `  x )  =  ~P ( J `  x ) )
1716ixpeq2dv 6834 . . . . . . 7  |-  ( ( h  =  H  /\  j  =  J )  -> 
X_ x  e.  ( s  X.  s ) ~P ( j `  x )  =  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
)
1814, 17eleq12d 2353 . . . . . 6  |-  ( ( h  =  H  /\  j  =  J )  ->  ( h  e.  X_ x  e.  ( s  X.  s ) ~P (
j `  x )  <->  H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x ) ) )
1918rexbidv 2566 . . . . 5  |-  ( ( h  =  H  /\  j  =  J )  ->  ( E. s  e. 
~P  t h  e.  X_ x  e.  (
s  X.  s ) ~P ( j `  x )  <->  E. s  e.  ~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) ) )
2013, 19anbi12d 691 . . . 4  |-  ( ( h  =  H  /\  j  =  J )  ->  ( ( j  Fn  ( t  X.  t
)  /\  E. s  e.  ~P  t h  e.  X_ x  e.  (
s  X.  s ) ~P ( j `  x ) )  <->  ( J  Fn  ( t  X.  t
)  /\  E. s  e.  ~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) ) ) )
2120exbidv 1614 . . 3  |-  ( ( h  =  H  /\  j  =  J )  ->  ( E. t ( j  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t
h  e.  X_ x  e.  ( s  X.  s
) ~P ( j `
 x ) )  <->  E. t ( J  Fn  ( t  X.  t
)  /\  E. s  e.  ~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) ) ) )
22 df-ssc 13689 . . 3  |-  C_cat  =  { <. h ,  j >.  |  E. t ( j  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t
h  e.  X_ x  e.  ( s  X.  s
) ~P ( j `
 x ) ) }
2321, 22brabga 4281 . 2  |-  ( ( H  e.  _V  /\  J  e.  _V )  ->  ( H  C_cat  J  <->  E. t
( J  Fn  (
t  X.  t )  /\  E. s  e. 
~P  t H  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) ) ) )
243, 11, 23pm5.21nii 342 1  |-  ( H 
C_cat  J  <->  E. t ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686   E.wrex 2546   _Vcvv 2790   ~Pcpw 3627   class class class wbr 4025    X. cxp 4689   Rel wrel 4696    Fn wfn 5252   ` cfv 5257   X_cixp 6819    C_cat cssc 13686
This theorem is referenced by:  sscpwex  13694  sscfn1  13696  sscfn2  13697  isssc  13699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ixp 6820  df-ssc 13689
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