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Theorem ralima 6061
Description: Universal 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
ralima  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( A. x  e.  ( F " B
) ph  <->  A. 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 ralima
StepHypRef Expression
1 fvex 5804 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 11 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  y  e.  B
)  ->  ( F `  y )  e.  _V )
3 fvelimab 5851 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  ( F `  y )  =  x ) )
4 eqcom 2461 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2861 . . 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, 8ralxfr2d 4611 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( A. x  e.  ( F " B
) ph  <->  A. y  e.  B  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797   _Vcvv 3072    C_ wss 3431   "cima 4946    Fn wfn 5516   ` cfv 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529
This theorem is referenced by:  supisolem  7826  ordtypelem6  7843  ordtypelem7  7844  limsupgle  13068  mrcuni  14673  ipodrsima  15449  mhmima  15605  ghmnsgima  15884  cntzmhm  15970  qtopeu  19416  kqdisj  19432  ghmcnp  19812  divstgplem  19818  qtopbaslem  20464  bndth  20657  fmcfil  20910  ovoliunlem1  21112  volsup2  21213  mbflimsup  21272  itg2gt0  21366  mdegleb  21663  efopn  22231  fsumdvdsmul  22663  imaelshi  25609  cvmopnlem  27306  ovoliunnfl  28576  voliunnfl  28578  volsupnfl  28579  gicabl  29597
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