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Theorem elabrexg 37343
Description: Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elabrexg  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Distinct variable groups:    x, A, y    y, B
Allowed substitution hints:    B( x)    V( x, y)

Proof of Theorem elabrexg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tru 1441 . . . . 5  |- T.
2 csbeq1a 3404 . . . . . . . 8  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
32equcoms 1849 . . . . . . 7  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4 a1tru 1453 . . . . . . 7  |-  ( z  =  x  -> T.  )
53, 42thd 243 . . . . . 6  |-  ( z  =  x  ->  ( B  =  [_ z  /  x ]_ B  <-> T.  )
)
65rspcev 3182 . . . . 5  |-  ( ( x  e.  A  /\ T.  )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
71, 6mpan2 675 . . . 4  |-  ( x  e.  A  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
87adantr 466 . . 3  |-  ( ( x  e.  A  /\  B  e.  V )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
9 eqeq1 2426 . . . . . 6  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
109rexbidv 2936 . . . . 5  |-  ( y  =  B  ->  ( E. z  e.  A  y  =  [_ z  /  x ]_ B  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
1110elabg 3218 . . . 4  |-  ( B  e.  V  ->  ( B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
1211adantl 467 . . 3  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( B  e.  {
y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
138, 12mpbird 235 . 2  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B } )
14 nfv 1755 . . . 4  |-  F/ z  y  =  B
15 nfcsb1v 3411 . . . . 5  |-  F/_ x [_ z  /  x ]_ B
1615nfeq2 2597 . . . 4  |-  F/ x  y  =  [_ z  /  x ]_ B
172eqeq2d 2436 . . . 4  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
1814, 16, 17cbvrex 3051 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. z  e.  A  y  =  [_ z  /  x ]_ B )
1918abbii 2551 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }
2013, 19syl6eleqr 2518 1  |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1872   {cab 2407   E.wrex 2772   [_csb 3395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-sbc 3300  df-csb 3396
This theorem is referenced by:  upbdrech  37477  ssfiunibd  37481
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