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

Theorem dmsnopg 5341
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmsnopg  |-  ( B  e.  V  ->  dom  {
<. A ,  B >. }  =  { A }
)

Proof of Theorem dmsnopg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2959 . . . . . 6  |-  x  e. 
_V
2 vex 2959 . . . . . 6  |-  y  e. 
_V
31, 2opth1 4434 . . . . 5  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  x  =  A )
43exlimiv 1644 . . . 4  |-  ( E. y <. x ,  y
>.  =  <. A ,  B >.  ->  x  =  A )
5 opeq1 3984 . . . . 5  |-  ( x  =  A  ->  <. x ,  B >.  =  <. A ,  B >. )
6 opeq2 3985 . . . . . . 7  |-  ( y  =  B  ->  <. x ,  y >.  =  <. x ,  B >. )
76eqeq1d 2444 . . . . . 6  |-  ( y  =  B  ->  ( <. x ,  y >.  =  <. A ,  B >.  <->  <. x ,  B >.  = 
<. A ,  B >. ) )
87spcegv 3037 . . . . 5  |-  ( B  e.  V  ->  ( <. x ,  B >.  = 
<. A ,  B >.  ->  E. y <. x ,  y
>.  =  <. A ,  B >. ) )
95, 8syl5 30 . . . 4  |-  ( B  e.  V  ->  (
x  =  A  ->  E. y <. x ,  y
>.  =  <. A ,  B >. ) )
104, 9impbid2 196 . . 3  |-  ( B  e.  V  ->  ( E. y <. x ,  y
>.  =  <. A ,  B >. 
<->  x  =  A ) )
111eldm2 5068 . . . 4  |-  ( x  e.  dom  { <. A ,  B >. }  <->  E. y <. x ,  y >.  e.  { <. A ,  B >. } )
12 opex 4427 . . . . . 6  |-  <. x ,  y >.  e.  _V
1312elsnc 3837 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. A ,  B >. }  <->  <. x ,  y
>.  =  <. A ,  B >. )
1413exbii 1592 . . . 4  |-  ( E. y <. x ,  y
>.  e.  { <. A ,  B >. }  <->  E. y <. x ,  y >.  =  <. A ,  B >. )
1511, 14bitri 241 . . 3  |-  ( x  e.  dom  { <. A ,  B >. }  <->  E. y <. x ,  y >.  =  <. A ,  B >. )
16 elsn 3829 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
1710, 15, 163bitr4g 280 . 2  |-  ( B  e.  V  ->  (
x  e.  dom  { <. A ,  B >. }  <-> 
x  e.  { A } ) )
1817eqrdv 2434 1  |-  ( B  e.  V  ->  dom  {
<. A ,  B >. }  =  { A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    = wceq 1652    e. wcel 1725   {csn 3814   <.cop 3817   dom cdm 4878
This theorem is referenced by:  dmsnopss  5342  dmpropg  5343  dmsnop  5344  rnsnopg  5349  fnsng  5498  funprg  5500  funtpg  5501  fntpg  5506  setsval  13493  eupap1  21698  bnj96  29236  bnj535  29261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-dm 4888
  Copyright terms: Public domain W3C validator