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Theorem dmsnopss 5343
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 5342 . . 3  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
2 eqimss 3401 . . 3  |-  ( dom 
{ <. A ,  B >. }  =  { A }  ->  dom  { <. A ,  B >. }  C_  { A } )
31, 2syl 16 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. } 
C_  { A }
)
4 opprc2 4008 . . . . . 6  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
54sneqd 3828 . . . . 5  |-  ( -.  B  e.  _V  ->  {
<. A ,  B >. }  =  { (/) } )
65dmeqd 5073 . . . 4  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  dom  { (/)
} )
7 dmsn0 5338 . . . 4  |-  dom  { (/)
}  =  (/)
86, 7syl6eq 2485 . . 3  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  (/) )
9 0ss 3657 . . 3  |-  (/)  C_  { A }
108, 9syl6eqss 3399 . 2  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  C_  { A } )
113, 10pm2.61i 159 1  |-  dom  { <. A ,  B >. } 
C_  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2957    C_ wss 3321   (/)c0 3629   {csn 3815   <.cop 3818   dom cdm 4879
This theorem is referenced by:  setsres  13496  setscom  13498  setsid  13509  strlemor1  13557  strle1  13561  constr3pthlem1  21643  ex-res  21750  funsnfsup  26744  mapfzcons1  26774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-dm 4889
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