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Theorem sess2 4790
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 3517 . . 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 3536 . . . . 5  |-  ( A 
C_  B  ->  { y  e.  A  |  y R x }  C_  { y  e.  B  | 
y R x }
)
3 ssexg 4539 . . . . . 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 434 . . . . 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 2829 . . 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 44 . 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 4781 . 2  |-  ( R Se  B  <->  A. x  e.  B  { y  e.  B  |  y R x }  e.  _V )
9 df-se 4781 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 270 1  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3071    C_ wss 3429   class class class wbr 4393   Se wse 4778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3073  df-in 3436  df-ss 3443  df-se 4781
This theorem is referenced by:  seeq2  4794  wereu2  4818  frmin  27840  wfrlem5  27865  frrlem5  27909
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