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Theorem ralima 5448
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 3004 . . . 4  |-  ( ( B  C_  A  /\  y  e.  B )  ->  y  e.  A )
2 funfvex 5244 . . . . 5  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
32funfni 5051 . . . 4  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
41, 3sylan2 280 . . 3  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  y  e.  B )
)  ->  ( F `  y )  e.  _V )
54anassrs 392 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  y  e.  B
)  ->  ( F `  y )  e.  _V )
6 fvelimab 5282 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( x  e.  ( F " B )  <->  E. y  e.  B  ( F `  y )  =  x ) )
7 eqcom 2085 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
87rexbii 2378 . . 3  |-  ( E. y  e.  B  ( F `  y )  =  x  <->  E. y  e.  B  x  =  ( F `  y ) )
96, 8syl6bb 194 . 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 271 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  x  =  ( F `  y ) )  ->  ( ph  <->  ps ) )
125, 9, 11ralxfr2d 4243 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   _Vcvv 2610    C_ wss 2983   "cima 4395    Fn wfn 4948   ` cfv 4953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-fv 4961
This theorem is referenced by:  supisolem  6516
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