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Theorem resopab2 4936
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 4933 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) }
2 ssel 3141 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
32pm4.71d 391 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  B ) ) )
43anbi1d 462 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  ph ) ) )
5 anass 399 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) )
64, 5bitr2di 196 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  ph )
) )
76opabbidv 4053 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  ( x  e.  B  /\  ph ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
81, 7eqtrid 2215 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 103    = wceq 1348    e. wcel 2141    C_ wss 3121   {copab 4047    |` cres 4611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-xp 4615  df-rel 4616  df-res 4621
This theorem is referenced by:  resmpt  4937
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