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Theorem resoprab2 5724
Description: Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
resoprab2  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }  |`  ( C  X.  D ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) } )
Distinct variable groups:    x, A, y, z    x, B, y, z    x, C, y, z    x, D, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem resoprab2
StepHypRef Expression
1 resoprab 5723 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  |`  ( C  X.  D ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) }
2 anass 393 . . . 4  |-  ( ( ( ( x  e.  C  /\  y  e.  D )  /\  (
x  e.  A  /\  y  e.  B )
)  /\  ph )  <->  ( (
x  e.  C  /\  y  e.  D )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) ) )
3 an4 553 . . . . . 6  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ( x  e.  A  /\  y  e.  B ) )  <->  ( (
x  e.  C  /\  x  e.  A )  /\  ( y  e.  D  /\  y  e.  B
) ) )
4 ssel 3017 . . . . . . . . 9  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
54pm4.71d 385 . . . . . . . 8  |-  ( C 
C_  A  ->  (
x  e.  C  <->  ( x  e.  C  /\  x  e.  A ) ) )
65bicomd 139 . . . . . . 7  |-  ( C 
C_  A  ->  (
( x  e.  C  /\  x  e.  A
)  <->  x  e.  C
) )
7 ssel 3017 . . . . . . . . 9  |-  ( D 
C_  B  ->  (
y  e.  D  -> 
y  e.  B ) )
87pm4.71d 385 . . . . . . . 8  |-  ( D 
C_  B  ->  (
y  e.  D  <->  ( y  e.  D  /\  y  e.  B ) ) )
98bicomd 139 . . . . . . 7  |-  ( D 
C_  B  ->  (
( y  e.  D  /\  y  e.  B
)  <->  y  e.  D
) )
106, 9bi2anan9 573 . . . . . 6  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( ( ( x  e.  C  /\  x  e.  A )  /\  (
y  e.  D  /\  y  e.  B )
)  <->  ( x  e.  C  /\  y  e.  D ) ) )
113, 10syl5bb 190 . . . . 5  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( ( ( x  e.  C  /\  y  e.  D )  /\  (
x  e.  A  /\  y  e.  B )
)  <->  ( x  e.  C  /\  y  e.  D ) ) )
1211anbi1d 453 . . . 4  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( ( ( ( x  e.  C  /\  y  e.  D )  /\  ( x  e.  A  /\  y  e.  B
) )  /\  ph ) 
<->  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) ) )
132, 12syl5bbr 192 . . 3  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( ( ( x  e.  C  /\  y  e.  D )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  ph ) )  <-> 
( ( x  e.  C  /\  y  e.  D )  /\  ph ) ) )
1413oprabbidv 5685 . 2  |-  ( ( C  C_  A  /\  D  C_  B )  ->  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  ph ) ) }  =  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
151, 14syl5eq 2132 1  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }  |`  ( C  X.  D ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2997    X. cxp 4426    |` cres 4430   {coprab 5635
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-rel 4435  df-res 4440  df-oprab 5638
This theorem is referenced by:  resmpt2  5725
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