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Theorem sess2 4819
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 3525 . . 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 3544 . . . . 5  |-  ( A 
C_  B  ->  { y  e.  A  |  y R x }  C_  { y  e.  B  | 
y R x }
)
3 ssexg 4567 . . . . . 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 435 . . . . 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 2835 . . 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 4810 . 2  |-  ( R Se  B  <->  A. x  e.  B  { y  e.  B  |  y R x }  e.  _V )
9 df-se 4810 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 273 1  |-  ( A 
C_  B  ->  ( R Se  B  ->  R Se  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1868   A.wral 2775   {crab 2779   _Vcvv 3081    C_ wss 3436   class class class wbr 4420   Se wse 4807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-in 3443  df-ss 3450  df-se 4810
This theorem is referenced by:  seeq2  4823  wereu2  4847  wfrlem5  7045  frmin  30475  frrlem5  30513
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