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Theorem ralrn 5558
Description: Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralrn  |-  ( F  Fn  A  ->  ( A. x  e.  ran  F
ph 
<-> 
A. y  e.  A  ps ) )
Distinct variable groups:    x, y, A   
x, F, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem ralrn
StepHypRef Expression
1 funfvex 5438 . . 3  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
21funfni 5223 . 2  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
3 fvelrnb 5469 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  x ) )
4 eqcom 2141 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2442 . . 3  |-  ( E. y  e.  A  ( F `  y )  =  x  <->  E. y  e.  A  x  =  ( F `  y ) )
63, 5syl6bb 195 . 2  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  x  =  ( F `  y ) ) )
7 rexrn.1 . . 3  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
87adantl 275 . 2  |-  ( ( F  Fn  A  /\  x  =  ( F `  y ) )  -> 
( ph  <->  ps ) )
92, 6, 8ralxfr2d 4385 1  |-  ( F  Fn  A  ->  ( A. x  e.  ran  F
ph 
<-> 
A. y  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   _Vcvv 2686   ran crn 4540    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-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  ralrnmpt  5562  cbvfo  5686  isoselem  5721  difinfsn  6985  nninfall  13234
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