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Theorem ralima 5657
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 ssel2 3092 . . . 4  |-  ( ( B  C_  A  /\  y  e.  B )  ->  y  e.  A )
2 funfvex 5438 . . . . 5  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
32funfni 5223 . . . 4  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
41, 3sylan2 284 . . 3  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  y  e.  B )
)  ->  ( F `  y )  e.  _V )
54anassrs 397 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  y  e.  B
)  ->  ( F `  y )  e.  _V )
6 fvelimab 5477 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  ( F `  y )  =  x ) )
7 eqcom 2141 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
87rexbii 2442 . . 3  |-  ( E. y  e.  B  ( F `  y )  =  x  <->  E. y  e.  B  x  =  ( F `  y ) )
96, 8syl6bb 195 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  x  =  ( F `  y ) ) )
10 rexima.x . . 3  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
1110adantl 275 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  x  =  ( F `  y ) )  ->  ( ph  <->  ps ) )
125, 9, 11ralxfr2d 4385 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( A. x  e.  ( F " B
) ph  <->  A. y  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   _Vcvv 2686    C_ wss 3071   "cima 4542    Fn wfn 5118   ` cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  supisolem  6895  qtopbasss  12706
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