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Theorem sess2 4519
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 3375 . . 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 3394 . . . . 5  |-  ( A 
C_  B  ->  { y  e.  A  |  y R x }  C_  { y  e.  B  | 
y R x }
)
3 ssexg 4317 . . . . . 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 424 . . . . 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 16 . . . 4  |-  ( A 
C_  B  ->  ( { y  e.  B  |  y R x }  e.  _V  ->  { y  e.  A  | 
y R x }  e.  _V ) )
65ralimdv 2753 . . 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 42 . 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 4510 . 2  |-  ( R Se  B  <->  A. x  e.  B  { y  e.  B  |  y R x }  e.  _V )
9 df-se 4510 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 262 1  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924    C_ wss 3288   class class class wbr 4180   Se wse 4507
This theorem is referenced by:  seeq2  4523  wereu2  4547  frmin  25464  wfrlem5  25482  frrlem5  25507
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rab 2683  df-v 2926  df-in 3295  df-ss 3302  df-se 4510
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