MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brab2a Structured version   Visualization version   Unicode version

Theorem brab2a 4906
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
Hypotheses
Ref Expression
brab2a.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brab2a.2  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
Assertion
Ref Expression
brab2a  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)

Proof of Theorem brab2a
StepHypRef Expression
1 simpl 463 . . . . 5  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ph )  -> 
( x  e.  C  /\  y  e.  D
) )
21ssopab2i 4746 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) }
3 brab2a.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
4 df-xp 4862 . . . 4  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
52, 3, 43sstr4i 3483 . . 3  |-  R  C_  ( C  X.  D
)
65brel 4905 . 2  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
7 df-br 4419 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
83eleq2i 2532 . . . 4  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
97, 8bitri 257 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
10 brab2a.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1110opelopab2a 4733 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
129, 11syl5bb 265 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ps ) )
136, 12biadan2 652 1  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   <.cop 3986   class class class wbr 4418   {copab 4476    X. cxp 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-xp 4862
This theorem is referenced by:  issect  15713  iscgrg  24613  ishlg  24703  iscgra  24907  isinag  24935  isleag  24939
  Copyright terms: Public domain W3C validator