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Theorem afvelrnb0 38811
Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5926. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
Assertion
Ref Expression
afvelrnb0  |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem afvelrnb0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrnafv 38809 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
21eleq2d 2534 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) } ) )
3 eqeq1 2475 . . . . . 6  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  B  =  ( F''' x ) ) )
4 eqcom 2478 . . . . . 6  |-  ( B  =  ( F''' x )  <-> 
( F''' x )  =  B )
53, 4syl6bb 269 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  ( F''' x )  =  B ) )
65rexbidv 2892 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F''' x )  <->  E. x  e.  A  ( F''' x )  =  B ) )
76elabg 3174 . . 3  |-  ( B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) }  ->  ( B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) }  <->  E. x  e.  A  ( F''' x )  =  B ) )
87ibi 249 . 2  |-  ( B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) }  ->  E. x  e.  A  ( F''' x )  =  B )
92, 8syl6bi 236 1  |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   {cab 2457   E.wrex 2757   ran crn 4840    Fn wfn 5584  '''cafv 38760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-dfat 38762  df-afv 38763
This theorem is referenced by:  ffnafv  38818
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