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Theorem fvelrnbf 36754
Description: A version of fvelrnb 5852 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 5852 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  B ) )
2 nfcv 2564 . . 3  |-  F/_ y A
3 fvelrnbf.1 . . 3  |-  F/_ x A
4 fvelrnbf.3 . . . . 5  |-  F/_ x F
5 nfcv 2564 . . . . 5  |-  F/_ x
y
64, 5nffv 5812 . . . 4  |-  F/_ x
( F `  y
)
7 fvelrnbf.2 . . . 4  |-  F/_ x B
86, 7nfeq 2575 . . 3  |-  F/ x
( F `  y
)  =  B
9 nfv 1728 . . 3  |-  F/ y ( F `  x
)  =  B
10 fveq2 5805 . . . 4  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1110eqeq1d 2404 . . 3  |-  ( y  =  x  ->  (
( F `  y
)  =  B  <->  ( F `  x )  =  B ) )
122, 3, 8, 9, 11cbvrexf 3028 . 2  |-  ( E. y  e.  A  ( F `  y )  =  B  <->  E. x  e.  A  ( F `  x )  =  B )
131, 12syl6bb 261 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 184    = wceq 1405    e. wcel 1842   F/_wnfc 2550   E.wrex 2754   ran crn 4943    Fn wfn 5520   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533
This theorem is referenced by:  refsumcn  36766  stoweidlem29  37161
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