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Theorem brssc 13942
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 13941 . . 3  |-  Rel  C_cat
2 brrelex12 4856 . . 3  |-  ( ( Rel  C_cat  /\  H  C_cat  J )  ->  ( H  e. 
_V  /\  J  e.  _V ) )
31, 2mpan 652 . 2  |-  ( H 
C_cat  J  ->  ( H  e.  _V  /\  J  e. 
_V ) )
4 vex 2903 . . . . . 6  |-  t  e. 
_V
54, 4xpex 4931 . . . . 5  |-  ( t  X.  t )  e. 
_V
6 fnex 5901 . . . . 5  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
75, 6mpan2 653 . . . 4  |-  ( J  Fn  ( t  X.  t )  ->  J  e.  _V )
8 elex 2908 . . . . 5  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  e.  _V )
98rexlimivw 2770 . . . 4  |-  ( E. s  e.  ~P  t H  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  ->  H  e.  _V )
107, 9anim12ci 551 . . 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 1641 . 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 448 . . . . . 6  |-  ( ( h  =  H  /\  j  =  J )  ->  j  =  J )
1312fneq1d 5477 . . . . 5  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j  Fn  (
t  X.  t )  <-> 
J  Fn  ( t  X.  t ) ) )
14 simpl 444 . . . . . . 7  |-  ( ( h  =  H  /\  j  =  J )  ->  h  =  H )
1512fveq1d 5671 . . . . . . . . 9  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j `  x
)  =  ( J `
 x ) )
1615pweqd 3748 . . . . . . . 8  |-  ( ( h  =  H  /\  j  =  J )  ->  ~P ( j `  x )  =  ~P ( J `  x ) )
1716ixpeq2dv 7015 . . . . . . 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 2456 . . . . . 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 2671 . . . . 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 692 . . . 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 1633 . . 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 13938 . . 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 4411 . 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 343 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   E.wrex 2651   _Vcvv 2900   ~Pcpw 3743   class class class wbr 4154    X. cxp 4817   Rel wrel 4824    Fn wfn 5390   ` cfv 5395   X_cixp 7000    C_cat cssc 13935
This theorem is referenced by:  sscpwex  13943  sscfn1  13945  sscfn2  13946  isssc  13948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ixp 7001  df-ssc 13938
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