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Theorem afvelrnb 28025
Description: A member of a function's range is a value of the function, analogous to fvelrnb 5570 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvelrnb  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F''' x )  =  B ) )
Distinct variable groups:    x, A    x, B    x, F
Allowed substitution hint:    V( x)

Proof of Theorem afvelrnb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrnafv 28024 . . . 4  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
21adantr 451 . . 3  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ran  F  =  {
y  |  E. x  e.  A  y  =  ( F''' x ) } )
32eleq2d 2350 . 2  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ( B  e.  ran  F  <-> 
B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) } ) )
4 eqeq1 2289 . . . . . 6  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  B  =  ( F''' x ) ) )
5 eqcom 2285 . . . . . 6  |-  ( B  =  ( F''' x )  <-> 
( F''' x )  =  B )
64, 5syl6bb 252 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  ( F''' x )  =  B ) )
76rexbidv 2564 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F''' x )  <->  E. x  e.  A  ( F''' x )  =  B ) )
87elabg 2915 . . 3  |-  ( B  e.  V  ->  ( B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) }  <->  E. x  e.  A  ( F''' x )  =  B ) )
98adantl 452 . 2  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ( B  e.  {
y  |  E. x  e.  A  y  =  ( F''' x ) }  <->  E. x  e.  A  ( F''' x )  =  B ) )
103, 9bitrd 244 1  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F''' x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   ran crn 4690    Fn wfn 5250  '''cafv 27972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-dfat 27974  df-afv 27975
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