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Theorem ffnafv 38763
Description: A function maps to a class to which all values belong, analogous to ffnfv 6032. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
ffnafv  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem ffnafv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ffn 5710 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fafvelrn 38762 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F''' x )  e.  B
)
32ralrimiva 2789 . . 3  |-  ( F : A --> B  ->  A. x  e.  A  ( F''' x )  e.  B
)
41, 3jca 539 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
) )
5 simpl 463 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F  Fn  A )
6 afvelrnb0 38756 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  y ) )
7 nfra1 2765 . . . . . 6  |-  F/ x A. x  e.  A  ( F''' x )  e.  B
8 nfv 1764 . . . . . 6  |-  F/ x  y  e.  B
9 rsp 2753 . . . . . . 7  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( F''' x )  e.  B
) )
10 eleq1 2517 . . . . . . . 8  |-  ( ( F''' x )  =  y  ->  ( ( F''' x )  e.  B  <->  y  e.  B ) )
1110biimpcd 232 . . . . . . 7  |-  ( ( F''' x )  e.  B  ->  ( ( F''' x )  =  y  ->  y  e.  B ) )
129, 11syl6 34 . . . . . 6  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( ( F''' x )  =  y  ->  y  e.  B ) ) )
137, 8, 12rexlimd 2846 . . . . 5  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( E. x  e.  A  ( F''' x )  =  y  ->  y  e.  B ) )
146, 13sylan9 667 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ( y  e.  ran  F  ->  y  e.  B ) )
1514ssrdv 3405 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ran  F  C_  B )
16 df-f 5564 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
175, 15, 16sylanbrc 675 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F : A
--> B )
184, 17impbii 192 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1447    e. wcel 1890   A.wral 2736   E.wrex 2737    C_ wss 3371   ran crn 4812    Fn wfn 5555   -->wf 5556  '''cafv 38705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-fv 5568  df-dfat 38707  df-afv 38708
This theorem is referenced by:  ffnaov  38791
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