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Theorem eldm3 27577
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 2986 . 2  |-  ( A  e.  dom  B  ->  A  e.  _V )
2 snprc 3944 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 reseq2 5110 . . . . 5  |-  ( { A }  =  (/)  ->  ( B  |`  { A } )  =  ( B  |`  (/) ) )
4 res0 5120 . . . . 5  |-  ( B  |`  (/) )  =  (/)
53, 4syl6eq 2491 . . . 4  |-  ( { A }  =  (/)  ->  ( B  |`  { A } )  =  (/) )
62, 5sylbi 195 . . 3  |-  ( -.  A  e.  _V  ->  ( B  |`  { A } )  =  (/) )
76necon1ai 2658 . 2  |-  ( ( B  |`  { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2503 . . 3  |-  ( x  =  A  ->  (
x  e.  dom  B  <->  A  e.  dom  B ) )
9 sneq 3892 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
109reseq2d 5115 . . . 4  |-  ( x  =  A  ->  ( B  |`  { x }
)  =  ( B  |`  { A } ) )
1110neeq1d 2626 . . 3  |-  ( x  =  A  ->  (
( B  |`  { x } )  =/=  (/)  <->  ( B  |` 
{ A } )  =/=  (/) ) )
12 df-clel 2439 . . . . 5  |-  ( <.
x ,  y >.  e.  B  <->  E. p ( p  =  <. x ,  y
>.  /\  p  e.  B
) )
1312exbii 1634 . . . 4  |-  ( E. y <. x ,  y
>.  e.  B  <->  E. y E. p ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
14 vex 2980 . . . . 5  |-  x  e. 
_V
1514eldm2 5043 . . . 4  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
16 n0 3651 . . . . 5  |-  ( ( B  |`  { x } )  =/=  (/)  <->  E. p  p  e.  ( B  |` 
{ x } ) )
17 elres 5150 . . . . . . 7  |-  ( p  e.  ( B  |`  { x } )  <->  E. z  e.  { x } E. y ( p  =  <. z ,  y
>.  /\  <. z ,  y
>.  e.  B ) )
18 eleq1 2503 . . . . . . . . . . 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 4064 . . . . . . . . . . . 12  |-  ( z  =  x  ->  <. z ,  y >.  =  <. x ,  y >. )
2120eqeq2d 2454 . . . . . . . . . . 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 1680 . . . . . . . 8  |-  ( z  =  x  ->  ( E. y ( p  = 
<. z ,  y >.  /\  <. z ,  y
>.  e.  B )  <->  E. y
( p  =  <. x ,  y >.  /\  p  e.  B ) ) )
2514, 24rexsn 3921 . . . . . . 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 1634 . . . . 5  |-  ( E. p  p  e.  ( B  |`  { x } )  <->  E. p E. y ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
28 excom 1787 . . . . 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 3036 . 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2611   E.wrex 2721   _Vcvv 2977   (/)c0 3642   {csn 3882   <.cop 3888   dom cdm 4845    |` cres 4847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-dm 4855  df-res 4857
This theorem is referenced by:  elrn3  27578
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