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Theorem ralima 5589
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 3042 . . . 4  |-  ( ( B  C_  A  /\  y  e.  B )  ->  y  e.  A )
2 funfvex 5370 . . . . 5  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
32funfni 5159 . . . 4  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
41, 3sylan2 282 . . 3  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  y  e.  B )
)  ->  ( F `  y )  e.  _V )
54anassrs 395 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  y  e.  B
)  ->  ( F `  y )  e.  _V )
6 fvelimab 5409 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  ( F `  y )  =  x ) )
7 eqcom 2102 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
87rexbii 2401 . . 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 273 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  x  =  ( F `  y ) )  ->  ( ph  <->  ps ) )
125, 9, 11ralxfr2d 4323 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 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   _Vcvv 2641    C_ wss 3021   "cima 4480    Fn wfn 5054   ` cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067
This theorem is referenced by:  supisolem  6810  qtopbasss  12443
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