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Theorem brssc 14006
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 14005 . . 3  |-  Rel  C_cat
2 brrelex12 4907 . . 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 2951 . . . . . 6  |-  t  e. 
_V
54, 4xpex 4982 . . . . 5  |-  ( t  X.  t )  e. 
_V
6 fnex 5953 . . . . 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 2956 . . . . 5  |-  ( H  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )  ->  H  e.  _V )
98rexlimivw 2818 . . . 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 1644 . 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 5528 . . . . 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 5722 . . . . . . . . 9  |-  ( ( h  =  H  /\  j  =  J )  ->  ( j `  x
)  =  ( J `
 x ) )
1615pweqd 3796 . . . . . . . 8  |-  ( ( h  =  H  /\  j  =  J )  ->  ~P ( j `  x )  =  ~P ( J `  x ) )
1716ixpeq2dv 7070 . . . . . . 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 2503 . . . . . 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 2718 . . . . 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 1636 . . 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 14002 . . 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 4461 . 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 1550    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   ~Pcpw 3791   class class class wbr 4204    X. cxp 4868   Rel wrel 4875    Fn wfn 5441   ` cfv 5446   X_cixp 7055    C_cat cssc 13999
This theorem is referenced by:  sscpwex  14007  sscfn1  14009  sscfn2  14010  isssc  14012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ixp 7056  df-ssc 14002
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