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Theorem ffnfv 5544
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 5240 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 ffvelrn 5519 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
32ralrimiva 2480 . . 3  |-  ( F : A --> B  ->  A. x  e.  A  ( F `  x )  e.  B )
41, 3jca 302 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
5 simpl 108 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F  Fn  A )
6 fvelrnb 5435 . . . . . 6  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  y ) )
76biimpd 143 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F `  x )  =  y ) )
8 nfra1 2441 . . . . . 6  |-  F/ x A. x  e.  A  ( F `  x )  e.  B
9 nfv 1491 . . . . . 6  |-  F/ x  y  e.  B
10 rsp 2455 . . . . . . 7  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  ( x  e.  A  ->  ( F `  x )  e.  B ) )
11 eleq1 2178 . . . . . . . 8  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  B  <->  y  e.  B ) )
1211biimpcd 158 . . . . . . 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 2521 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  ( E. x  e.  A  ( F `  x )  =  y  ->  y  e.  B ) )
157, 14sylan9 404 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  -> 
( y  e.  ran  F  ->  y  e.  B
) )
1615ssrdv 3071 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
17 df-f 5095 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
185, 16, 17sylanbrc 411 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  F : A --> B )
194, 18impbii 125 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392    C_ wss 3039   ran crn 4508    Fn wfn 5086   -->wf 5087   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099
This theorem is referenced by:  ffnfvf  5545  fnfvrnss  5546  fmpt2d  5548  ffnov  5841  elixpconst  6566  elixpsn  6595  ctssdccl  6962  cnref1o  9389  shftf  10542  eff2  11285  reeff1  11306  dvfre  12726  isomninnlem  13036
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