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

Theorem opelco 5011
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 4396 . 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 5010 . 2  |-  ( A ( C  o.  D
) B  <->  E. x
( A D x  /\  x C B ) )
51, 4bitr3i 259 1  |-  ( <. A ,  B >.  e.  ( C  o.  D
)  <->  E. x ( A D x  /\  x C B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   E.wex 1671    e. wcel 1904   _Vcvv 3031   <.cop 3965   class class class wbr 4395    o. ccom 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-co 4848
This theorem is referenced by:  dmcoss  5100  dmcosseq  5102  cotrg  5217  coiun  5352  co02  5356  coi1  5358  coass  5361  fmptco  6072  dftpos4  7010  fmptcof2  28334  cnvco1  30471  cnvco2  30472  txpss3v  30716  dffun10  30752  coiun1  36315
  Copyright terms: Public domain W3C validator