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Theorem afvelrnb0 30211
Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5841. (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 30209 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
21eleq2d 2521 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) } ) )
3 eqeq1 2455 . . . . . 6  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  B  =  ( F''' x ) ) )
4 eqcom 2460 . . . . . 6  |-  ( B  =  ( F''' x )  <-> 
( F''' x )  =  B )
53, 4syl6bb 261 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F''' x )  <->  ( F''' x )  =  B ) )
65rexbidv 2855 . . . 4  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F''' x )  <->  E. x  e.  A  ( F''' x )  =  B ) )
76elabg 3207 . . 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 241 . 2  |-  ( B  e.  { y  |  E. x  e.  A  y  =  ( F''' x ) }  ->  E. x  e.  A  ( F''' x )  =  B )
92, 8syl6bi 228 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 1370    e. wcel 1758   {cab 2436   E.wrex 2796   ran crn 4942    Fn wfn 5514  '''cafv 30159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527  df-dfat 30161  df-afv 30162
This theorem is referenced by:  ffnafv  30218
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