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Theorem dmsnopss 5463
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 5462 . . 3  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
2 eqimss 3541 . . 3  |-  ( dom 
{ <. A ,  B >. }  =  { A }  ->  dom  { <. A ,  B >. }  C_  { A } )
31, 2syl 16 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. } 
C_  { A }
)
4 opprc2 4227 . . . . . 6  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
54sneqd 4028 . . . . 5  |-  ( -.  B  e.  _V  ->  {
<. A ,  B >. }  =  { (/) } )
65dmeqd 5194 . . . 4  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  dom  { (/)
} )
7 dmsn0 5458 . . . 4  |-  dom  { (/)
}  =  (/)
86, 7syl6eq 2511 . . 3  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  (/) )
9 0ss 3813 . . 3  |-  (/)  C_  { A }
108, 9syl6eqss 3539 . 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 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   (/)c0 3783   {csn 4016   <.cop 4022   dom cdm 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998
This theorem is referenced by:  snopsuppss  6906  setsres  14746  setscom  14748  setsid  14759  strlemor1  14811  strle1  14815  funsnfsupOLD  18451  constr3pthlem1  24857  ex-res  25364  mapfzcons1  30889
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