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

Theorem ffvresb 5900
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 5595 . . . . . 6  |-  ( ( F  |`  A ) : A --> B  ->  dom  ( F  |`  A )  =  A )
2 dmres 5167 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
3 inss2 3562 . . . . . . 7  |-  ( A  i^i  dom  F )  C_ 
dom  F
42, 3eqsstri 3378 . . . . . 6  |-  dom  ( F  |`  A )  C_  dom  F
51, 4syl6eqssr 3399 . . . . 5  |-  ( ( F  |`  A ) : A --> B  ->  A  C_ 
dom  F )
65sselda 3348 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  x  e.  dom  F
)
7 fvres 5745 . . . . . 6  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
87adantl 453 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
9 ffvelrn 5868 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  e.  B )
108, 9eqeltrrd 2511 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
116, 10jca 519 . . 3  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) )
1211ralrimiva 2789 . 2  |-  ( ( F  |`  A ) : A --> B  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) )
13 simpl 444 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  x  e.  dom  F )
1413ralimi 2781 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  x  e.  dom  F )
15 dfss3 3338 . . . . . 6  |-  ( A 
C_  dom  F  <->  A. x  e.  A  x  e.  dom  F )
1614, 15sylibr 204 . . . . 5  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A  C_  dom  F )
17 funfn 5482 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
18 fnssres 5558 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  C_  dom  F
)  ->  ( F  |`  A )  Fn  A
)
1917, 18sylanb 459 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A )  Fn  A )
2016, 19sylan2 461 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A )  Fn  A
)
21 simpr 448 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F `  x )  e.  B
)
227eleq1d 2502 . . . . . . . 8  |-  ( x  e.  A  ->  (
( ( F  |`  A ) `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
2321, 22syl5ibr 213 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  e.  dom  F  /\  ( F `  x )  e.  B
)  ->  ( ( F  |`  A ) `  x )  e.  B
) )
2423ralimia 2779 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  ( ( F  |`  A ) `  x
)  e.  B )
2524adantl 453 . . . . 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 5896 . . . . 5  |-  ( ( ( F  |`  A )  Fn  A  /\  A. x  e.  A  (
( F  |`  A ) `
 x )  e.  B )  ->  ran  ( F  |`  A ) 
C_  B )
2720, 25, 26syl2anc 643 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ran  ( F  |`  A )  C_  B
)
28 df-f 5458 . . . 4  |-  ( ( F  |`  A ) : A --> B  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  B ) )
2920, 27, 28sylanbrc 646 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A ) : A --> B )
3029ex 424 . 2  |-  ( Fun 
F  ->  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F  |`  A ) : A --> B ) )
3112, 30impbid2 196 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320   dom cdm 4878   ran crn 4879    |` cres 4880   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454
This theorem is referenced by:  lmbr2  17323  lmff  17365  lmmbr2  19212  iscau2  19230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462
  Copyright terms: Public domain W3C validator