MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sess2 Unicode version

Theorem sess2 4362
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 3237 . . 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 3256 . . . . 5  |-  ( A 
C_  B  ->  { y  e.  A  |  y R x }  C_  { y  e.  B  | 
y R x }
)
3 ssexg 4160 . . . . . 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 423 . . . . 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 15 . . . 4  |-  ( A 
C_  B  ->  ( { y  e.  B  |  y R x }  e.  _V  ->  { y  e.  A  | 
y R x }  e.  _V ) )
65ralimdv 2622 . . 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 40 . 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 4353 . 2  |-  ( R Se  B  <->  A. x  e.  B  { y  e.  B  |  y R x }  e.  _V )
9 df-se 4353 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 261 1  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   Se wse 4350
This theorem is referenced by:  seeq2  4366  wereu2  4390  frmin  24242  wfrlem5  24260  frrlem5  24285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-se 4353
  Copyright terms: Public domain W3C validator