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Theorem rexima 5954
Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypothesis
Ref Expression
rexima.x  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexima  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. x  e.  ( F " B
) ph  <->  E. y  e.  B  ps ) )
Distinct variable groups:    ph, y    ps, x    x, F, y    x, B, y    x, A, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem rexima
StepHypRef Expression
1 fvex 5699 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 11 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  y  e.  B
)  ->  ( F `  y )  e.  _V )
3 fvelimab 5745 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  ( F `  y )  =  x ) )
4 eqcom 2443 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2738 . . 3  |-  ( E. y  e.  B  ( F `  y )  =  x  <->  E. y  e.  B  x  =  ( F `  y ) )
63, 5syl6bb 261 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  x  =  ( F `  y ) ) )
7 rexima.x . . 3  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
87adantl 466 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  x  =  ( F `  y ) )  ->  ( ph  <->  ps ) )
92, 6, 8rexxfr2d 4507 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. x  e.  ( F " B
) ph  <->  E. y  e.  B  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714   _Vcvv 2970    C_ wss 3326   "cima 4841    Fn wfn 5411   ` cfv 5416
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 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
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-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-fv 5424
This theorem is referenced by:  supisolem  7718  ipodrsima  15333  lmflf  19576  caucfil  20792  dyadmbllem  21077  lhop1lem  21483  mblfinlem1  28425  itg2gt0cn  28444
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