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Theorem fvelrnbf 27689
Description: A version of fvelrnb 5570 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 5570 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  B ) )
2 nfcv 2419 . . 3  |-  F/_ y A
3 fvelrnbf.1 . . 3  |-  F/_ x A
4 fvelrnbf.3 . . . . 5  |-  F/_ x F
5 nfcv 2419 . . . . 5  |-  F/_ x
y
64, 5nffv 5532 . . . 4  |-  F/_ x
( F `  y
)
7 fvelrnbf.2 . . . 4  |-  F/_ x B
86, 7nfeq 2426 . . 3  |-  F/ x
( F `  y
)  =  B
9 nfv 1605 . . 3  |-  F/ y ( F `  x
)  =  B
10 fveq2 5525 . . . 4  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1110eqeq1d 2291 . . 3  |-  ( y  =  x  ->  (
( F `  y
)  =  B  <->  ( F `  x )  =  B ) )
122, 3, 8, 9, 11cbvrexf 2759 . 2  |-  ( E. y  e.  A  ( F `  y )  =  B  <->  E. x  e.  A  ( F `  x )  =  B )
131, 12syl6bb 252 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 176    = wceq 1623    e. wcel 1684   F/_wnfc 2406   E.wrex 2544   ran crn 4690    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  refsumcn  27701  stoweidlem29  27778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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