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Theorem resopab2 4705
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 4702 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) }
2 ssel 3002 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
32pm4.71d 385 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  B ) ) )
43anbi1d 453 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  ph ) ) )
5 anass 393 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) )
64, 5syl6rbb 195 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  ph )
) )
76opabbidv 3864 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  ( x  e.  B  /\  ph ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
81, 7syl5eq 2127 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 102    = wceq 1285    e. wcel 1434    C_ wss 2982   {copab 3858    |` cres 4393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-xp 4397  df-rel 4398  df-res 4403
This theorem is referenced by:  resmpt  4706
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