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

Theorem opelco 5122
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
opelco.1  |-  A  e. 
_V
opelco.2  |-  B  e. 
_V
Assertion
Ref Expression
opelco  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  E. x ( A D x  /\  x C B ) )
Distinct variable groups:    x, A    x, B    x, C    x, D

Proof of Theorem opelco
StepHypRef Expression
1 df-br 4404 . 2  |-  ( A ( C  o.  D
) B  <->  <. A ,  B >.  e.  ( C  o.  D ) )
2 opelco.1 . . 3  |-  A  e. 
_V
3 opelco.2 . . 3  |-  B  e. 
_V
42, 3brco 5121 . 2  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
51, 4bitr3i 251 1  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  E. x ( A D x  /\  x C B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   _Vcvv 3078   <.cop 3994   class class class wbr 4403    o. ccom 4955
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-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  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-br 4404  df-opab 4462  df-co 4960
This theorem is referenced by:  dmcoss  5210  dmcosseq  5212  cotr  5321  coiun  5458  co02  5462  coi1  5464  coass  5467  fmptco  5988  dftpos4  6877  fmptcof2  26150  cnvco1  27734  cnvco2  27735  txpss3v  28073  dffun10  28109  tfrqfree  28146
  Copyright terms: Public domain W3C validator