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Theorem resoprab2 5918
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 5917 . 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 399 . . . 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 576 . . . . . 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 3122 . . . . . . . . 9  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
54pm4.71d 391 . . . . . . . 8  |-  ( C 
C_  A  ->  (
x  e.  C  <->  ( x  e.  C  /\  x  e.  A ) ) )
65bicomd 140 . . . . . . 7  |-  ( C 
C_  A  ->  (
( x  e.  C  /\  x  e.  A
)  <->  x  e.  C
) )
7 ssel 3122 . . . . . . . . 9  |-  ( D 
C_  B  ->  (
y  e.  D  -> 
y  e.  B ) )
87pm4.71d 391 . . . . . . . 8  |-  ( D 
C_  B  ->  (
y  e.  D  <->  ( y  e.  D  /\  y  e.  B ) ) )
98bicomd 140 . . . . . . 7  |-  ( D 
C_  B  ->  (
( y  e.  D  /\  y  e.  B
)  <->  y  e.  D
) )
106, 9bi2anan9 596 . . . . . 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 191 . . . . 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 461 . . . 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, 12bitr3id 193 . . 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 5875 . 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 2202 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 103    = wceq 1335    e. wcel 2128    C_ wss 3102    X. cxp 4584    |` cres 4588   {coprab 5825
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4592  df-rel 4593  df-res 4598  df-oprab 5828
This theorem is referenced by:  resmpo  5919
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