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Theorem ffnafv 30077
Description: A function maps to a class to which all values belong, analogous to ffnfv 5869. (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 5559 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fafvelrn 30076 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F''' x )  e.  B
)
32ralrimiva 2799 . . 3  |-  ( F : A --> B  ->  A. x  e.  A  ( F''' x )  e.  B
)
41, 3jca 532 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
) )
5 simpl 457 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F  Fn  A )
6 afvelrnb0 30070 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  y ) )
7 nfra1 2766 . . . . . 6  |-  F/ x A. x  e.  A  ( F''' x )  e.  B
8 nfv 1673 . . . . . 6  |-  F/ x  y  e.  B
9 rsp 2776 . . . . . . 7  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( F''' x )  e.  B
) )
10 eleq1 2503 . . . . . . . 8  |-  ( ( F''' x )  =  y  ->  ( ( F''' x )  e.  B  <->  y  e.  B ) )
1110biimpcd 224 . . . . . . 7  |-  ( ( F''' x )  e.  B  ->  ( ( F''' x )  =  y  ->  y  e.  B ) )
129, 11syl6 33 . . . . . 6  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( ( F''' x )  =  y  ->  y  e.  B ) ) )
137, 8, 12rexlimd 2838 . . . . 5  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( E. x  e.  A  ( F''' x )  =  y  ->  y  e.  B ) )
146, 13sylan9 657 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ( y  e.  ran  F  ->  y  e.  B ) )
1514ssrdv 3362 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ran  F  C_  B )
16 df-f 5422 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
175, 15, 16sylanbrc 664 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F : A
--> B )
184, 17impbii 188 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    C_ wss 3328   ran crn 4841    Fn wfn 5413   -->wf 5414  '''cafv 30018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-dfat 30020  df-afv 30021
This theorem is referenced by:  ffnaov  30105
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