MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffvresb Structured version   Unicode version

Theorem ffvresb 6069
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 5750 . . . . . 6  |-  ( ( F  |`  A ) : A --> B  ->  dom  ( F  |`  A )  =  A )
2 dmres 5144 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
3 inss2 3683 . . . . . . 7  |-  ( A  i^i  dom  F )  C_ 
dom  F
42, 3eqsstri 3494 . . . . . 6  |-  dom  ( F  |`  A )  C_  dom  F
51, 4syl6eqssr 3515 . . . . 5  |-  ( ( F  |`  A ) : A --> B  ->  A  C_ 
dom  F )
65sselda 3464 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  x  e.  dom  F
)
7 fvres 5895 . . . . . 6  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
87adantl 467 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
9 ffvelrn 6035 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  e.  B )
108, 9eqeltrrd 2508 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
116, 10jca 534 . . 3  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) )
1211ralrimiva 2836 . 2  |-  ( ( F  |`  A ) : A --> B  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) )
13 simpl 458 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  x  e.  dom  F )
1413ralimi 2815 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  x  e.  dom  F )
15 dfss3 3454 . . . . . 6  |-  ( A 
C_  dom  F  <->  A. x  e.  A  x  e.  dom  F )
1614, 15sylibr 215 . . . . 5  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A  C_  dom  F )
17 funfn 5630 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
18 fnssres 5707 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  C_  dom  F
)  ->  ( F  |`  A )  Fn  A
)
1917, 18sylanb 474 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A )  Fn  A )
2016, 19sylan2 476 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A )  Fn  A
)
21 simpr 462 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F `  x )  e.  B
)
227eleq1d 2491 . . . . . . . 8  |-  ( x  e.  A  ->  (
( ( F  |`  A ) `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
2321, 22syl5ibr 224 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  e.  dom  F  /\  ( F `  x )  e.  B
)  ->  ( ( F  |`  A ) `  x )  e.  B
) )
2423ralimia 2813 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  ( ( F  |`  A ) `  x
)  e.  B )
2524adantl 467 . . . . 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 6066 . . . . 5  |-  ( ( ( F  |`  A )  Fn  A  /\  A. x  e.  A  (
( F  |`  A ) `
 x )  e.  B )  ->  ran  ( F  |`  A ) 
C_  B )
2720, 25, 26syl2anc 665 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ran  ( F  |`  A )  C_  B
)
28 df-f 5605 . . . 4  |-  ( ( F  |`  A ) : A --> B  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  B ) )
2920, 27, 28sylanbrc 668 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A ) : A --> B )
3029ex 435 . 2  |-  ( Fun 
F  ->  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F  |`  A ) : A --> B ) )
3112, 30impbid2 207 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771    i^i cin 3435    C_ wss 3436   dom cdm 4853   ran crn 4854    |` cres 4855   Fun wfun 5595    Fn wfn 5596   -->wf 5597   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609
This theorem is referenced by:  lmbr2  20273  lmff  20315  lmmbr2  22227  iscau2  22245  relogbf  23726  sseqf  29233  fourierdlem97  38007
  Copyright terms: Public domain W3C validator