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Theorem fvelrnbf 27666
Description: A version of fvelrnb 5775 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 5775 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  B ) )
2 nfcv 2573 . . 3  |-  F/_ y A
3 fvelrnbf.1 . . 3  |-  F/_ x A
4 fvelrnbf.3 . . . . 5  |-  F/_ x F
5 nfcv 2573 . . . . 5  |-  F/_ x
y
64, 5nffv 5736 . . . 4  |-  F/_ x
( F `  y
)
7 fvelrnbf.2 . . . 4  |-  F/_ x B
86, 7nfeq 2580 . . 3  |-  F/ x
( F `  y
)  =  B
9 nfv 1630 . . 3  |-  F/ y ( F `  x
)  =  B
10 fveq2 5729 . . . 4  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1110eqeq1d 2445 . . 3  |-  ( y  =  x  ->  (
( F `  y
)  =  B  <->  ( F `  x )  =  B ) )
122, 3, 8, 9, 11cbvrexf 2928 . 2  |-  ( E. y  e.  A  ( F `  y )  =  B  <->  E. x  e.  A  ( F `  x )  =  B )
131, 12syl6bb 254 1  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   F/_wnfc 2560   E.wrex 2707   ran crn 4880    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  refsumcn  27678  stoweidlem29  27755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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