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Theorem eldm3 29118
Description: Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
Assertion
Ref Expression
eldm3  |-  ( A  e.  dom  B  <->  ( B  |` 
{ A } )  =/=  (/) )

Proof of Theorem eldm3
Dummy variables  x  y  z  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2  |-  ( A  e.  dom  B  ->  A  e.  _V )
2 snprc 4097 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 reseq2 5274 . . . . 5  |-  ( { A }  =  (/)  ->  ( B  |`  { A } )  =  ( B  |`  (/) ) )
4 res0 5284 . . . . 5  |-  ( B  |`  (/) )  =  (/)
53, 4syl6eq 2524 . . . 4  |-  ( { A }  =  (/)  ->  ( B  |`  { A } )  =  (/) )
62, 5sylbi 195 . . 3  |-  ( -.  A  e.  _V  ->  ( B  |`  { A } )  =  (/) )
76necon1ai 2698 . 2  |-  ( ( B  |`  { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2539 . . 3  |-  ( x  =  A  ->  (
x  e.  dom  B  <->  A  e.  dom  B ) )
9 sneq 4043 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
109reseq2d 5279 . . . 4  |-  ( x  =  A  ->  ( B  |`  { x }
)  =  ( B  |`  { A } ) )
1110neeq1d 2744 . . 3  |-  ( x  =  A  ->  (
( B  |`  { x } )  =/=  (/)  <->  ( B  |` 
{ A } )  =/=  (/) ) )
12 df-clel 2462 . . . . 5  |-  ( <.
x ,  y >.  e.  B  <->  E. p ( p  =  <. x ,  y
>.  /\  p  e.  B
) )
1312exbii 1644 . . . 4  |-  ( E. y <. x ,  y
>.  e.  B  <->  E. y E. p ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
14 vex 3121 . . . . 5  |-  x  e. 
_V
1514eldm2 5207 . . . 4  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
16 n0 3799 . . . . 5  |-  ( ( B  |`  { x } )  =/=  (/)  <->  E. p  p  e.  ( B  |` 
{ x } ) )
17 elres 5315 . . . . . . 7  |-  ( p  e.  ( B  |`  { x } )  <->  E. z  e.  { x } E. y ( p  =  <. z ,  y
>.  /\  <. z ,  y
>.  e.  B ) )
18 eleq1 2539 . . . . . . . . . . 11  |-  ( p  =  <. z ,  y
>.  ->  ( p  e.  B  <->  <. z ,  y
>.  e.  B ) )
1918pm5.32i 637 . . . . . . . . . 10  |-  ( ( p  =  <. z ,  y >.  /\  p  e.  B )  <->  ( p  =  <. z ,  y
>.  /\  <. z ,  y
>.  e.  B ) )
20 opeq1 4219 . . . . . . . . . . . 12  |-  ( z  =  x  ->  <. z ,  y >.  =  <. x ,  y >. )
2120eqeq2d 2481 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
p  =  <. z ,  y >.  <->  p  =  <. x ,  y >.
) )
2221anbi1d 704 . . . . . . . . . 10  |-  ( z  =  x  ->  (
( p  =  <. z ,  y >.  /\  p  e.  B )  <->  ( p  =  <. x ,  y
>.  /\  p  e.  B
) ) )
2319, 22syl5bbr 259 . . . . . . . . 9  |-  ( z  =  x  ->  (
( p  =  <. z ,  y >.  /\  <. z ,  y >.  e.  B
)  <->  ( p  = 
<. x ,  y >.  /\  p  e.  B
) ) )
2423exbidv 1690 . . . . . . . 8  |-  ( z  =  x  ->  ( E. y ( p  = 
<. z ,  y >.  /\  <. z ,  y
>.  e.  B )  <->  E. y
( p  =  <. x ,  y >.  /\  p  e.  B ) ) )
2514, 24rexsn 4073 . . . . . . 7  |-  ( E. z  e.  { x } E. y ( p  =  <. z ,  y
>.  /\  <. z ,  y
>.  e.  B )  <->  E. y
( p  =  <. x ,  y >.  /\  p  e.  B ) )
2617, 25bitri 249 . . . . . 6  |-  ( p  e.  ( B  |`  { x } )  <->  E. y ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
2726exbii 1644 . . . . 5  |-  ( E. p  p  e.  ( B  |`  { x } )  <->  E. p E. y ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
28 excom 1798 . . . . 5  |-  ( E. p E. y ( p  =  <. x ,  y >.  /\  p  e.  B )  <->  E. y E. p ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
2916, 27, 283bitri 271 . . . 4  |-  ( ( B  |`  { x } )  =/=  (/)  <->  E. y E. p ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
3013, 15, 293bitr4i 277 . . 3  |-  ( x  e.  dom  B  <->  ( B  |` 
{ x } )  =/=  (/) )
318, 11, 30vtoclbg 3177 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  ( B  |` 
{ A } )  =/=  (/) ) )
321, 7, 31pm5.21nii 353 1  |-  ( A  e.  dom  B  <->  ( B  |` 
{ A } )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118   (/)c0 3790   {csn 4033   <.cop 4039   dom cdm 5005    |` cres 5007
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 4574  ax-nul 4582  ax-pr 4692
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-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-dm 5015  df-res 5017
This theorem is referenced by:  elrn3  29119
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