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Theorem sess1 4552
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess1  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )

Proof of Theorem sess1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 445 . . . . . 6  |-  ( ( R  C_  S  /\  y  e.  A )  ->  R  C_  S )
21ssbrd 4255 . . . . 5  |-  ( ( R  C_  S  /\  y  e.  A )  ->  ( y R x  ->  y S x ) )
32ss2rabdv 3426 . . . 4  |-  ( R 
C_  S  ->  { y  e.  A  |  y R x }  C_  { y  e.  A  | 
y S x }
)
4 ssexg 4351 . . . . 5  |-  ( ( { y  e.  A  |  y R x }  C_  { y  e.  A  |  y S x }  /\  { y  e.  A  | 
y S x }  e.  _V )  ->  { y  e.  A  |  y R x }  e.  _V )
54ex 425 . . . 4  |-  ( { y  e.  A  | 
y R x }  C_ 
{ y  e.  A  |  y S x }  ->  ( {
y  e.  A  | 
y S x }  e.  _V  ->  { y  e.  A  |  y R x }  e.  _V ) )
63, 5syl 16 . . 3  |-  ( R 
C_  S  ->  ( { y  e.  A  |  y S x }  e.  _V  ->  { y  e.  A  | 
y R x }  e.  _V ) )
76ralimdv 2787 . 2  |-  ( R 
C_  S  ->  ( A. x  e.  A  { y  e.  A  |  y S x }  e.  _V  ->  A. x  e.  A  {
y  e.  A  | 
y R x }  e.  _V ) )
8 df-se 4544 . 2  |-  ( S Se  A  <->  A. x  e.  A  { y  e.  A  |  y S x }  e.  _V )
9 df-se 4544 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 263 1  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    C_ wss 3322   class class class wbr 4214   Se wse 4541
This theorem is referenced by:  seeq1  4556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-in 3329  df-ss 3336  df-br 4215  df-se 4544
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