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Theorem ffnafv 31723
Description: A function maps to a class to which all values belong, analogous to ffnfv 6045. (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 5729 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fafvelrn 31722 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F''' x )  e.  B
)
32ralrimiva 2878 . . 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 31716 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  y ) )
7 nfra1 2845 . . . . . 6  |-  F/ x A. x  e.  A  ( F''' x )  e.  B
8 nfv 1683 . . . . . 6  |-  F/ x  y  e.  B
9 rsp 2830 . . . . . . 7  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( F''' x )  e.  B
) )
10 eleq1 2539 . . . . . . . 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 2947 . . . . 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 3510 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ran  F  C_  B )
16 df-f 5590 . . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ran crn 5000    Fn wfn 5581   -->wf 5582  '''cafv 31666
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-res 5011  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-dfat 31668  df-afv 31669
This theorem is referenced by:  ffnaov  31751
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