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Theorem sess1 4684
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 454 . . . . . 6  |-  ( ( R  C_  S  /\  y  e.  A )  ->  R  C_  S )
21ssbrd 4330 . . . . 5  |-  ( ( R  C_  S  /\  y  e.  A )  ->  ( y R x  ->  y S x ) )
32ss2rabdv 3430 . . . 4  |-  ( R 
C_  S  ->  { y  e.  A  |  y R x }  C_  { y  e.  A  | 
y S x }
)
4 ssexg 4435 . . . . 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 434 . . . 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 2793 . 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 4676 . 2  |-  ( S Se  A  <->  A. x  e.  A  { y  e.  A  |  y S x }  e.  _V )
9 df-se 4676 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 270 1  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1761   A.wral 2713   {crab 2717   _Vcvv 2970    C_ wss 3325   class class class wbr 4289   Se wse 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-v 2972  df-in 3332  df-ss 3339  df-br 4290  df-se 4676
This theorem is referenced by:  seeq1  4688
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