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Theorem ffvresb 5551
Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
ffvresb  |-  ( Fun 
F  ->  ( ( F  |`  A ) : A --> B  <->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem ffvresb
StepHypRef Expression
1 fdm 5248 . . . . . 6  |-  ( ( F  |`  A ) : A --> B  ->  dom  ( F  |`  A )  =  A )
2 dmres 4810 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
3 inss2 3267 . . . . . . 7  |-  ( A  i^i  dom  F )  C_ 
dom  F
42, 3eqsstri 3099 . . . . . 6  |-  dom  ( F  |`  A )  C_  dom  F
51, 4eqsstrrdi 3120 . . . . 5  |-  ( ( F  |`  A ) : A --> B  ->  A  C_ 
dom  F )
65sselda 3067 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  x  e.  dom  F
)
7 fvres 5413 . . . . . 6  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
87adantl 275 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
9 ffvelrn 5521 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  e.  B )
108, 9eqeltrrd 2195 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
116, 10jca 304 . . 3  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) )
1211ralrimiva 2482 . 2  |-  ( ( F  |`  A ) : A --> B  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) )
13 simpl 108 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  x  e.  dom  F )
1413ralimi 2472 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  x  e.  dom  F )
15 dfss3 3057 . . . . . 6  |-  ( A 
C_  dom  F  <->  A. x  e.  A  x  e.  dom  F )
1614, 15sylibr 133 . . . . 5  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A  C_  dom  F )
17 funfn 5123 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
18 fnssres 5206 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  C_  dom  F
)  ->  ( F  |`  A )  Fn  A
)
1917, 18sylanb 282 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A )  Fn  A )
2016, 19sylan2 284 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A )  Fn  A
)
21 simpr 109 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F `  x )  e.  B
)
227eleq1d 2186 . . . . . . . 8  |-  ( x  e.  A  ->  (
( ( F  |`  A ) `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
2321, 22syl5ibr 155 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  e.  dom  F  /\  ( F `  x )  e.  B
)  ->  ( ( F  |`  A ) `  x )  e.  B
) )
2423ralimia 2470 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  ( ( F  |`  A ) `  x
)  e.  B )
2524adantl 275 . . . . 5  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  A. x  e.  A  ( ( F  |`  A ) `  x )  e.  B
)
26 fnfvrnss 5548 . . . . 5  |-  ( ( ( F  |`  A )  Fn  A  /\  A. x  e.  A  (
( F  |`  A ) `
 x )  e.  B )  ->  ran  ( F  |`  A ) 
C_  B )
2720, 25, 26syl2anc 408 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ran  ( F  |`  A )  C_  B
)
28 df-f 5097 . . . 4  |-  ( ( F  |`  A ) : A --> B  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  B ) )
2920, 27, 28sylanbrc 413 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A ) : A --> B )
3029ex 114 . 2  |-  ( Fun 
F  ->  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F  |`  A ) : A --> B ) )
3112, 30impbid2 142 1  |-  ( Fun 
F  ->  ( ( F  |`  A ) : A --> B  <->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393    i^i cin 3040    C_ wss 3041   dom cdm 4509   ran crn 4510    |` cres 4511   Fun wfun 5087    Fn wfn 5088   -->wf 5089   ` cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  resflem  5552  tfrcl  6229  frecfcllem  6269  lmbr2  12294  lmff  12329
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