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Theorem resoprab2 6100
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 6099 . 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 631 . . . 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 798 . . . . . 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 3279 . . . . . . . . 9  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
54pm4.71d 616 . . . . . . . 8  |-  ( C 
C_  A  ->  (
x  e.  C  <->  ( x  e.  C  /\  x  e.  A ) ) )
65bicomd 193 . . . . . . 7  |-  ( C 
C_  A  ->  (
( x  e.  C  /\  x  e.  A
)  <->  x  e.  C
) )
7 ssel 3279 . . . . . . . . 9  |-  ( D 
C_  B  ->  (
y  e.  D  -> 
y  e.  B ) )
87pm4.71d 616 . . . . . . . 8  |-  ( D 
C_  B  ->  (
y  e.  D  <->  ( y  e.  D  /\  y  e.  B ) ) )
98bicomd 193 . . . . . . 7  |-  ( D 
C_  B  ->  (
( y  e.  D  /\  y  e.  B
)  <->  y  e.  D
) )
106, 9bi2anan9 844 . . . . . 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 249 . . . . 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 686 . . . 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 251 . . 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 6061 . 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 2425 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 359    = wceq 1649    e. wcel 1717    C_ wss 3257    X. cxp 4810    |` cres 4814   {coprab 6015
This theorem is referenced by:  resmpt2  6101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-opab 4202  df-xp 4818  df-rel 4819  df-res 4824  df-oprab 6018
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