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Theorem ffvresb 5964
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 5643 . . . . . 6  |-  ( ( F  |`  A ) : A --> B  ->  dom  ( F  |`  A )  =  A )
2 dmres 5206 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
3 inss2 3633 . . . . . . 7  |-  ( A  i^i  dom  F )  C_ 
dom  F
42, 3eqsstri 3447 . . . . . 6  |-  dom  ( F  |`  A )  C_  dom  F
51, 4syl6eqssr 3468 . . . . 5  |-  ( ( F  |`  A ) : A --> B  ->  A  C_ 
dom  F )
65sselda 3417 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  x  e.  dom  F
)
7 fvres 5788 . . . . . 6  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
87adantl 464 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
9 ffvelrn 5931 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  e.  B )
108, 9eqeltrrd 2471 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
116, 10jca 530 . . 3  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) )
1211ralrimiva 2796 . 2  |-  ( ( F  |`  A ) : A --> B  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) )
13 simpl 455 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  x  e.  dom  F )
1413ralimi 2775 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  x  e.  dom  F )
15 dfss3 3407 . . . . . 6  |-  ( A 
C_  dom  F  <->  A. x  e.  A  x  e.  dom  F )
1614, 15sylibr 212 . . . . 5  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A  C_  dom  F )
17 funfn 5525 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
18 fnssres 5602 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  C_  dom  F
)  ->  ( F  |`  A )  Fn  A
)
1917, 18sylanb 470 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A )  Fn  A )
2016, 19sylan2 472 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A )  Fn  A
)
21 simpr 459 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F `  x )  e.  B
)
227eleq1d 2451 . . . . . . . 8  |-  ( x  e.  A  ->  (
( ( F  |`  A ) `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
2321, 22syl5ibr 221 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  e.  dom  F  /\  ( F `  x )  e.  B
)  ->  ( ( F  |`  A ) `  x )  e.  B
) )
2423ralimia 2773 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  ( ( F  |`  A ) `  x
)  e.  B )
2524adantl 464 . . . . 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 5961 . . . . 5  |-  ( ( ( F  |`  A )  Fn  A  /\  A. x  e.  A  (
( F  |`  A ) `
 x )  e.  B )  ->  ran  ( F  |`  A ) 
C_  B )
2720, 25, 26syl2anc 659 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ran  ( F  |`  A )  C_  B
)
28 df-f 5500 . . . 4  |-  ( ( F  |`  A ) : A --> B  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  B ) )
2920, 27, 28sylanbrc 662 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A ) : A --> B )
3029ex 432 . 2  |-  ( Fun 
F  ->  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F  |`  A ) : A --> B ) )
3112, 30impbid2 204 1  |-  ( Fun 
F  ->  ( ( F  |`  A ) : A --> B  <->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732    i^i cin 3388    C_ wss 3389   dom cdm 4913   ran crn 4914    |` cres 4915   Fun wfun 5490    Fn wfn 5491   -->wf 5492   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504
This theorem is referenced by:  lmbr2  19846  lmff  19888  lmmbr2  21783  iscau2  21801  relogbf  23249  sseqf  28514  fourierdlem97  32152
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