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Theorem resopab2 4861
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 4858 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) }
2 ssel 3086 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
32pm4.71d 390 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  B ) ) )
43anbi1d 460 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  ph ) ) )
5 anass 398 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) )
64, 5syl6rbb 196 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  ph )
) )
76opabbidv 3989 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  ( x  e.  B  /\  ph ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
81, 7syl5eq 2182 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 1331    e. wcel 1480    C_ wss 3066   {copab 3983    |` cres 4536
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540  df-rel 4541  df-res 4546
This theorem is referenced by:  resmpt  4862
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