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Theorem dmsnopss 5161
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 5160 . . 3  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
2 eqimss 3243 . . 3  |-  ( dom 
{ <. A ,  B >. }  =  { A }  ->  dom  { <. A ,  B >. }  C_  { A } )
31, 2syl 15 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. } 
C_  { A }
)
4 opprc2 3835 . . . . . 6  |-  ( -.  B  e.  _V  ->  <. A ,  B >.  =  (/) )
54sneqd 3666 . . . . 5  |-  ( -.  B  e.  _V  ->  {
<. A ,  B >. }  =  { (/) } )
65dmeqd 4897 . . . 4  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  dom  { (/)
} )
7 dmsn0 5156 . . . 4  |-  dom  { (/)
}  =  (/)
86, 7syl6eq 2344 . . 3  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  =  (/) )
9 0ss 3496 . . . 4  |-  (/)  C_  { A }
109a1i 10 . . 3  |-  ( -.  B  e.  _V  ->  (/)  C_ 
{ A } )
118, 10eqsstrd 3225 . 2  |-  ( -.  B  e.  _V  ->  dom 
{ <. A ,  B >. }  C_  { A } )
123, 11pm2.61i 156 1  |-  dom  { <. A ,  B >. } 
C_  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   dom cdm 4705
This theorem is referenced by:  setsres  13190  setscom  13192  setsid  13203  strlemor1  13251  strle1  13255  ex-res  20844  fvsnn  25217  funsnfsup  26865  mapfzcons1  26897  constr3pthlem1  28401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715
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