ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resoprab2 Unicode version

Theorem resoprab2 5950
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 5949 . 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 581 . . . . . 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 3141 . . . . . . . . 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 3141 . . . . . . . . 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 601 . . . . . 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 462 . . . 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 5907 . 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, 14eqtrid 2215 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 1348    e. wcel 2141    C_ wss 3121    X. cxp 4609    |` cres 4613   {coprab 5854
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-res 4623  df-oprab 5857
This theorem is referenced by:  resmpo  5951
  Copyright terms: Public domain W3C validator