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Theorem dmsnopss 5478
Description: The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on  B). (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
dmsnopss  |-  dom  { <. A ,  B >. } 
C_  { A }

Proof of Theorem dmsnopss
StepHypRef Expression
1 dmsnopg 5477 . . 3  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
2 eqimss 3556 . . 3  |-  ( dom 
{ <. A ,  B >. }  =  { A }  ->  dom  { <. A ,  B >. }  C_  { A } )
31, 2syl 16 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. } 
C_  { A }
)
4 opprc2 4237 . . . . . 6  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
54sneqd 4039 . . . . 5  |-  ( -.  B  e.  _V  ->  {
<. A ,  B >. }  =  { (/) } )
65dmeqd 5203 . . . 4  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  dom  { (/)
} )
7 dmsn0 5473 . . . 4  |-  dom  { (/)
}  =  (/)
86, 7syl6eq 2524 . . 3  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  (/) )
9 0ss 3814 . . 3  |-  (/)  C_  { A }
108, 9syl6eqss 3554 . 2  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  C_  { A } )
113, 10pm2.61i 164 1  |-  dom  { <. A ,  B >. } 
C_  { A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009
This theorem is referenced by:  snopsuppss  6913  setsres  14514  setscom  14516  setsid  14527  strlemor1  14578  strle1  14582  funsnfsupOLD  18027  constr3pthlem1  24331  ex-res  24839  mapfzcons1  30253
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