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Theorem resoprab2 6394
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 6393 . 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 649 . . . 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 822 . . . . . 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 3503 . . . . . . . . 9  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
54pm4.71d 634 . . . . . . . 8  |-  ( C 
C_  A  ->  (
x  e.  C  <->  ( x  e.  C  /\  x  e.  A ) ) )
65bicomd 201 . . . . . . 7  |-  ( C 
C_  A  ->  (
( x  e.  C  /\  x  e.  A
)  <->  x  e.  C
) )
7 ssel 3503 . . . . . . . . 9  |-  ( D 
C_  B  ->  (
y  e.  D  -> 
y  e.  B ) )
87pm4.71d 634 . . . . . . . 8  |-  ( D 
C_  B  ->  (
y  e.  D  <->  ( y  e.  D  /\  y  e.  B ) ) )
98bicomd 201 . . . . . . 7  |-  ( D 
C_  B  ->  (
( y  e.  D  /\  y  e.  B
)  <->  y  e.  D
) )
106, 9bi2anan9 871 . . . . . 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 257 . . . . 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 704 . . . 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 259 . . 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 6346 . 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 2520 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481    X. cxp 5003    |` cres 5007   {coprab 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-xp 5011  df-rel 5012  df-res 5017  df-oprab 6299
This theorem is referenced by:  resmpt2  6395
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