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Theorem eldm3 29366
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 3118 . 2  |-  ( A  e.  dom  B  ->  A  e.  _V )
2 snprc 4095 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 reseq2 5278 . . . . 5  |-  ( { A }  =  (/)  ->  ( B  |`  { A } )  =  ( B  |`  (/) ) )
4 res0 5288 . . . . 5  |-  ( B  |`  (/) )  =  (/)
53, 4syl6eq 2514 . . . 4  |-  ( { A }  =  (/)  ->  ( B  |`  { A } )  =  (/) )
62, 5sylbi 195 . . 3  |-  ( -.  A  e.  _V  ->  ( B  |`  { A } )  =  (/) )
76necon1ai 2688 . 2  |-  ( ( B  |`  { A } )  =/=  (/)  ->  A  e.  _V )
8 eleq1 2529 . . 3  |-  ( x  =  A  ->  (
x  e.  dom  B  <->  A  e.  dom  B ) )
9 sneq 4042 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
109reseq2d 5283 . . . 4  |-  ( x  =  A  ->  ( B  |`  { x }
)  =  ( B  |`  { A } ) )
1110neeq1d 2734 . . 3  |-  ( x  =  A  ->  (
( B  |`  { x } )  =/=  (/)  <->  ( B  |` 
{ A } )  =/=  (/) ) )
12 df-clel 2452 . . . . 5  |-  ( <.
x ,  y >.  e.  B  <->  E. p ( p  =  <. x ,  y
>.  /\  p  e.  B
) )
1312exbii 1668 . . . 4  |-  ( E. y <. x ,  y
>.  e.  B  <->  E. y E. p ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
14 vex 3112 . . . . 5  |-  x  e. 
_V
1514eldm2 5211 . . . 4  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
16 n0 3803 . . . . 5  |-  ( ( B  |`  { x } )  =/=  (/)  <->  E. p  p  e.  ( B  |` 
{ x } ) )
17 elres 5319 . . . . . . 7  |-  ( p  e.  ( B  |`  { x } )  <->  E. z  e.  { x } E. y ( p  =  <. z ,  y
>.  /\  <. z ,  y
>.  e.  B ) )
18 eleq1 2529 . . . . . . . . . . 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 2471 . . . . . . . . . . 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 1715 . . . . . . . 8  |-  ( z  =  x  ->  ( E. y ( p  = 
<. z ,  y >.  /\  <. z ,  y
>.  e.  B )  <->  E. y
( p  =  <. x ,  y >.  /\  p  e.  B ) ) )
2514, 24rexsn 4072 . . . . . . 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 1668 . . . . 5  |-  ( E. p  p  e.  ( B  |`  { x } )  <->  E. p E. y ( p  = 
<. x ,  y >.  /\  p  e.  B
) )
28 excom 1850 . . . . 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 3168 . 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109   (/)c0 3793   {csn 4032   <.cop 4038   dom cdm 5008    |` cres 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-dm 5018  df-res 5020
This theorem is referenced by:  elrn3  29367
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