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Theorem resopab2 5266
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
Assertion
Ref Expression
resopab2  |-  ( A 
C_  B  ->  ( { <. x ,  y
>.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem resopab2
StepHypRef Expression
1 resopab 5264 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) }
2 ssel 3461 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
32pm4.71d 634 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  B ) ) )
43anbi1d 704 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  ph ) ) )
5 anass 649 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( x  e.  A  /\  (
x  e.  B  /\  ph ) ) )
64, 5syl6rbb 262 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ( x  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  ph )
) )
76opabbidv 4466 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  ( x  e.  B  /\  ph ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
81, 7syl5eq 2507 1  |-  ( A 
C_  B  ->  ( { <. x ,  y
>.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3439   {copab 4460    |` cres 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-opab 4462  df-xp 4957  df-rel 4958  df-res 4963
This theorem is referenced by:  resmpt  5267  marypha2lem4  7802
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