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

Theorem opeldm 4995
Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
Hypotheses
Ref Expression
opeldm.1  |-  A  e. 
_V
opeldm.2  |-  B  e. 
_V
Assertion
Ref Expression
opeldm  |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )

Proof of Theorem opeldm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 opeldm.2 . . 3  |-  B  e. 
_V
2 opeq2 4126 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
32eleq1d 2485 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
41, 3spcev 3111 . 2  |-  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C )
5 opeldm.1 . . 3  |-  A  e. 
_V
65eldm2 4990 . 2  |-  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C )
74, 6sylibr 215 1  |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3017   <.cop 3942   dom cdm 4791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-rab 2718  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-br 4362  df-dm 4801
This theorem is referenced by:  breldm  4996  elreldm  5016  relssres  5099  iss  5109  imadmrn  5135  dfco2a  5292  funssres  5579  funun  5581  tz7.48-1  7110  iiner  7385  r0weon  8390  axdc3lem2  8827  uzrdgfni  12117  imasaddfnlem  15372  imasvscafn  15381  cicsym  15647  gsum2d  17542  dfcnv2  28220  bnj1379  29589
  Copyright terms: Public domain W3C validator