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Theorem fvelrnbf 30971
Description: A version of fvelrnb 5913 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 5913 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  B ) )
2 nfcv 2629 . . 3  |-  F/_ y A
3 fvelrnbf.1 . . 3  |-  F/_ x A
4 fvelrnbf.3 . . . . 5  |-  F/_ x F
5 nfcv 2629 . . . . 5  |-  F/_ x
y
64, 5nffv 5871 . . . 4  |-  F/_ x
( F `  y
)
7 fvelrnbf.2 . . . 4  |-  F/_ x B
86, 7nfeq 2640 . . 3  |-  F/ x
( F `  y
)  =  B
9 nfv 1683 . . 3  |-  F/ y ( F `  x
)  =  B
10 fveq2 5864 . . . 4  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1110eqeq1d 2469 . . 3  |-  ( y  =  x  ->  (
( F `  y
)  =  B  <->  ( F `  x )  =  B ) )
122, 3, 8, 9, 11cbvrexf 3083 . 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 1379    e. wcel 1767   F/_wnfc 2615   E.wrex 2815   ran crn 5000    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  refsumcn  30983  stoweidlem29  31329
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