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Theorem ffnfv 5456
Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
Assertion
Ref Expression
ffnfv  |-  ( 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 ffnfv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ffn 5161 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 ffvelrn 5432 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
32ralrimiva 2446 . . 3  |-  ( F : A --> B  ->  A. x  e.  A  ( F `  x )  e.  B )
41, 3jca 300 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
5 simpl 107 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F  Fn  A )
6 fvelrnb 5352 . . . . . 6  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
76biimpd 142 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
8 nfra1 2409 . . . . . 6  |-  F/ x A. x  e.  A  ( F `  x )  e.  B
9 nfv 1466 . . . . . 6  |-  F/ x  y  e.  B
10 rsp 2423 . . . . . . 7  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  ( x  e.  A  ->  ( F `  x )  e.  B ) )
11 eleq1 2150 . . . . . . . 8  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  B  <->  y  e.  B ) )
1211biimpcd 157 . . . . . . 7  |-  ( ( F `  x )  e.  B  ->  (
( F `  x
)  =  y  -> 
y  e.  B ) )
1310, 12syl6 33 . . . . . 6  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  ( x  e.  A  ->  (
( F `  x
)  =  y  -> 
y  e.  B ) ) )
148, 9, 13rexlimd 2486 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  ( E. x  e.  A  ( F `  x )  =  y  ->  y  e.  B ) )
157, 14sylan9 401 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  -> 
( y  e.  ran  F  ->  y  e.  B
) )
1615ssrdv 3031 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
17 df-f 5019 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
185, 16, 17sylanbrc 408 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> B )
194, 18impbii 124 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360    C_ wss 2999   ran crn 4439    Fn wfn 5010   -->wf 5011   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023
This theorem is referenced by:  ffnfvf  5457  fnfvrnss  5458  fmpt2d  5460  ffnov  5749  cnref1o  9131  shftf  10260  eff2  10966  reeff1  10987
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