MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resopab2 Unicode version

Theorem resopab2 4999
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
Assertion
Ref Expression
resopab2  |-  ( A 
C_  B  ->  ( { <. x ,  y
>.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem resopab2
StepHypRef Expression
1 resopab 4996 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) }
2 ssel 3174 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
32pm4.71d 615 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  B ) ) )
43anbi1d 685 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  ph ) ) )
5 anass 630 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) )
64, 5syl6rbb 253 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  ph )
) )
76opabbidv 4082 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  ( x  e.  B  /\  ph ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
81, 7syl5eq 2327 1  |-  ( A 
C_  B  ->  ( { <. x ,  y
>.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {copab 4076    |` cres 4691
This theorem is referenced by:  resmpt  5000  marypha2lem4  7191
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-res 4701
  Copyright terms: Public domain W3C validator