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Theorem fvelrnbf 37339
Description: A version of fvelrnb 5912 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
fvelrnbf.1  |-  F/_ x A
fvelrnbf.2  |-  F/_ x B
fvelrnbf.3  |-  F/_ x F
Assertion
Ref Expression
fvelrnbf  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )

Proof of Theorem fvelrnbf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvelrnb 5912 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  B ) )
2 nfcv 2592 . . 3  |-  F/_ y A
3 fvelrnbf.1 . . 3  |-  F/_ x A
4 fvelrnbf.3 . . . . 5  |-  F/_ x F
5 nfcv 2592 . . . . 5  |-  F/_ x
y
64, 5nffv 5872 . . . 4  |-  F/_ x
( F `  y
)
7 fvelrnbf.2 . . . 4  |-  F/_ x B
86, 7nfeq 2603 . . 3  |-  F/ x
( F `  y
)  =  B
9 nfv 1761 . . 3  |-  F/ y ( F `  x
)  =  B
10 fveq2 5865 . . . 4  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1110eqeq1d 2453 . . 3  |-  ( y  =  x  ->  (
( F `  y
)  =  B  <->  ( F `  x )  =  B ) )
122, 3, 8, 9, 11cbvrexf 3014 . 2  |-  ( E. y  e.  A  ( F `  y )  =  B  <->  E. x  e.  A  ( F `  x )  =  B )
131, 12syl6bb 265 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    <-> wb 188    = wceq 1444    e. wcel 1887   F/_wnfc 2579   E.wrex 2738   ran crn 4835    Fn wfn 5577   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590
This theorem is referenced by:  refsumcn  37351  stoweidlem29  37889  stoweidlem29OLD  37890
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