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Theorem sess2 4806
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess2  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )

Proof of Theorem sess2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3495 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  { y  e.  B  |  y R x }  e.  _V  ->  A. x  e.  A  {
y  e.  B  | 
y R x }  e.  _V ) )
2 rabss2 3514 . . . . 5  |-  ( A 
C_  B  ->  { y  e.  A  |  y R x }  C_  { y  e.  B  | 
y R x }
)
3 ssexg 4552 . . . . . 6  |-  ( ( { y  e.  A  |  y R x }  C_  { y  e.  B  |  y R x }  /\  { y  e.  B  | 
y R x }  e.  _V )  ->  { y  e.  A  |  y R x }  e.  _V )
43ex 436 . . . . 5  |-  ( { y  e.  A  | 
y R x }  C_ 
{ y  e.  B  |  y R x }  ->  ( {
y  e.  B  | 
y R x }  e.  _V  ->  { y  e.  A  |  y R x }  e.  _V ) )
52, 4syl 17 . . . 4  |-  ( A 
C_  B  ->  ( { y  e.  B  |  y R x }  e.  _V  ->  { y  e.  A  | 
y R x }  e.  _V ) )
65ralimdv 2800 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  A  { y  e.  B  |  y R x }  e.  _V  ->  A. x  e.  A  {
y  e.  A  | 
y R x }  e.  _V ) )
71, 6syld 45 . 2  |-  ( A 
C_  B  ->  ( A. x  e.  B  { y  e.  B  |  y R x }  e.  _V  ->  A. x  e.  A  {
y  e.  A  | 
y R x }  e.  _V ) )
8 df-se 4797 . 2  |-  ( R Se  B  <->  A. x  e.  B  { y  e.  B  |  y R x }  e.  _V )
9 df-se 4797 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 274 1  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1889   A.wral 2739   {crab 2743   _Vcvv 3047    C_ wss 3406   class class class wbr 4405   Se wse 4794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rab 2748  df-v 3049  df-in 3413  df-ss 3420  df-se 4797
This theorem is referenced by:  seeq2  4810  wereu2  4834  wfrlem5  7045  frmin  30492  frrlem5  30530
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