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Theorem brssc 14019
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 14018 . . 3  |-  Rel  C_cat
2 brrelex12 4918 . . 3  |-  ( ( Rel  C_cat  /\  H  C_cat  J )  ->  ( H  e. 
_V  /\  J  e.  _V ) )
31, 2mpan 653 . 2  |-  ( H 
C_cat  J  ->  ( H  e.  _V  /\  J  e. 
_V ) )
4 vex 2961 . . . . . 6  |-  t  e. 
_V
54, 4xpex 4993 . . . . 5  |-  ( t  X.  t )  e. 
_V
6 fnex 5964 . . . . 5  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
75, 6mpan2 654 . . . 4  |-  ( J  Fn  ( t  X.  t )  ->  J  e.  _V )
8 elex 2966 . . . . 5  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  e.  _V )
98rexlimivw 2828 . . . 4  |-  ( E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  ->  H  e.  _V )
107, 9anim12ci 552 . . 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 1645 . 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 449 . . . . . 6  |-  ( ( h  =  H  /\  j  =  J )  ->  j  =  J )
1312fneq1d 5539 . . . . 5  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j  Fn  (
t  X.  t )  <-> 
J  Fn  ( t  X.  t ) ) )
14 simpl 445 . . . . . . 7  |-  ( ( h  =  H  /\  j  =  J )  ->  h  =  H )
1512fveq1d 5733 . . . . . . . . 9  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j `  x
)  =  ( J `
 x ) )
1615pweqd 3806 . . . . . . . 8  |-  ( ( h  =  H  /\  j  =  J )  ->  ~P ( j `  x )  =  ~P ( J `  x ) )
1716ixpeq2dv 7081 . . . . . . 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 2506 . . . . . 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 2728 . . . . 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 693 . . . 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 1637 . . 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 14015 . . 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 4472 . 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 344 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958   ~Pcpw 3801   class class class wbr 4215    X. cxp 4879   Rel wrel 4886    Fn wfn 5452   ` cfv 5457   X_cixp 7066    C_cat cssc 14012
This theorem is referenced by:  sscpwex  14020  sscfn1  14022  sscfn2  14023  isssc  14025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ixp 7067  df-ssc 14015
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