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

Theorem ffvresb 5690
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 5393 . . . . . 6  |-  ( ( F  |`  A ) : A --> B  ->  dom  ( F  |`  A )  =  A )
2 dmres 4976 . . . . . . . 8  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
3 inss2 3390 . . . . . . . 8  |-  ( A  i^i  dom  F )  C_ 
dom  F
42, 3eqsstri 3208 . . . . . . 7  |-  dom  ( F  |`  A )  C_  dom  F
54a1i 10 . . . . . 6  |-  ( ( F  |`  A ) : A --> B  ->  dom  ( F  |`  A ) 
C_  dom  F )
61, 5eqsstr3d 3213 . . . . 5  |-  ( ( F  |`  A ) : A --> B  ->  A  C_ 
dom  F )
76sselda 3180 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  x  e.  dom  F
)
8 fvres 5542 . . . . . 6  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
98adantl 452 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  =  ( F `
 x ) )
10 ffvelrn 5663 . . . . 5  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( ( F  |`  A ) `  x
)  e.  B )
119, 10eqeltrrd 2358 . . . 4  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
127, 11jca 518 . . 3  |-  ( ( ( F  |`  A ) : A --> B  /\  x  e.  A )  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) )
1312ralrimiva 2626 . 2  |-  ( ( F  |`  A ) : A --> B  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  B ) )
14 simpl 443 . . . . . . 7  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  x  e.  dom  F )
1514ralimi 2618 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  x  e.  dom  F )
16 dfss3 3170 . . . . . 6  |-  ( A 
C_  dom  F  <->  A. x  e.  A  x  e.  dom  F )
1715, 16sylibr 203 . . . . 5  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A  C_  dom  F )
18 funfn 5283 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
19 fnssres 5357 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  C_  dom  F
)  ->  ( F  |`  A )  Fn  A
)
2018, 19sylanb 458 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A )  Fn  A )
2117, 20sylan2 460 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A )  Fn  A
)
22 simpr 447 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F `  x )  e.  B
)
238eleq1d 2349 . . . . . . . 8  |-  ( x  e.  A  ->  (
( ( F  |`  A ) `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
2422, 23syl5ibr 212 . . . . . . 7  |-  ( x  e.  A  ->  (
( x  e.  dom  F  /\  ( F `  x )  e.  B
)  ->  ( ( F  |`  A ) `  x )  e.  B
) )
2524ralimia 2616 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  A. x  e.  A  ( ( F  |`  A ) `  x
)  e.  B )
2625adantl 452 . . . . 5  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  A. x  e.  A  ( ( F  |`  A ) `  x )  e.  B
)
27 fnfvrnss 5687 . . . . 5  |-  ( ( ( F  |`  A )  Fn  A  /\  A. x  e.  A  (
( F  |`  A ) `
 x )  e.  B )  ->  ran  ( F  |`  A ) 
C_  B )
2821, 26, 27syl2anc 642 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ran  ( F  |`  A )  C_  B
)
29 df-f 5259 . . . 4  |-  ( ( F  |`  A ) : A --> B  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  B ) )
3021, 28, 29sylanbrc 645 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) )  ->  ( F  |`  A ) : A --> B )
3130ex 423 . 2  |-  ( Fun 
F  ->  ( A. x  e.  A  (
x  e.  dom  F  /\  ( F `  x
)  e.  B )  ->  ( F  |`  A ) : A --> B ) )
3213, 31impbid2 195 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  lmbr2  16989  lmff  17029  lmmbr2  18685  iscau2  18703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
  Copyright terms: Public domain W3C validator