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Theorem afvelrnb 27994
Description: A member of a function's range is a value of the function, analogous to fvelrnb 5766 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 27993 . . . 4  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
21adantr 452 . . 3  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ran  F  =  {
y  |  E. x  e.  A  y  =  ( F''' x ) } )
32eleq2d 2502 . 2  |-  ( ( F  Fn  A  /\  B  e.  V )  ->  ( B  e.  ran  F  <-> 
B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) } ) )
4 eqeq1 2441 . . . . . 6  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  B  =  ( F''' x ) ) )
5 eqcom 2437 . . . . . 6  |-  ( B  =  ( F''' x )  <-> 
( F''' x )  =  B )
64, 5syl6bb 253 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  ( F''' x )  =  B ) )
76rexbidv 2718 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F''' x )  <->  E. x  e.  A  ( F''' x )  =  B ) )
87elabg 3075 . . 3  |-  ( B  e.  V  ->  ( B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) }  <->  E. x  e.  A  ( F''' x )  =  B ) )
98adantl 453 . 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 245 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   ran crn 4871    Fn wfn 5441  '''cafv 27939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-dfat 27941  df-afv 27942
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