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

Theorem funfvima 5950
Description: A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
Assertion
Ref Expression
funfvima  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )

Proof of Theorem funfvima
StepHypRef Expression
1 dmres 5129 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21elin2 3539 . . . . . 6  |-  ( B  e.  dom  ( F  |`  A )  <->  ( B  e.  A  /\  B  e. 
dom  F ) )
3 funres 5455 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
4 fvelrn 5837 . . . . . . . . 9  |-  ( ( Fun  ( F  |`  A )  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
53, 4sylan 471 . . . . . . . 8  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
6 fvres 5702 . . . . . . . . . 10  |-  ( B  e.  A  ->  (
( F  |`  A ) `
 B )  =  ( F `  B
) )
76eleq1d 2507 . . . . . . . . 9  |-  ( B  e.  A  ->  (
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) ) )
8 df-ima 4851 . . . . . . . . . 10  |-  ( F
" A )  =  ran  ( F  |`  A )
98eleq2i 2505 . . . . . . . . 9  |-  ( ( F `  B )  e.  ( F " A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) )
107, 9syl6rbbr 264 . . . . . . . 8  |-  ( B  e.  A  ->  (
( F `  B
)  e.  ( F
" A )  <->  ( ( F  |`  A ) `  B )  e.  ran  ( F  |`  A ) ) )
115, 10syl5ibrcom 222 . . . . . . 7  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
1211ex 434 . . . . . 6  |-  ( Fun 
F  ->  ( B  e.  dom  ( F  |`  A )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
132, 12syl5bir 218 . . . . 5  |-  ( Fun 
F  ->  ( ( B  e.  A  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1413expd 436 . . . 4  |-  ( Fun 
F  ->  ( B  e.  A  ->  ( B  e.  dom  F  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) ) ) )
1514com12 31 . . 3  |-  ( B  e.  A  ->  ( Fun  F  ->  ( B  e.  dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) ) )
1615impd 431 . 2  |-  ( B  e.  A  ->  (
( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1716pm2.43b 50 1  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   dom cdm 4838   ran crn 4839    |` cres 4840   "cima 4841   Fun wfun 5410   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-fv 5424
This theorem is referenced by:  funfvima2  5951  tz7.48-2  6895  tz9.12lem3  7994  lindff1  18247  txcnp  19191  c1liplem1  21466  htthlem  24317  elovimad  25954  tpr2rico  26340  brsiga  26595  erdszelem8  27084  nobndlem2  27832  nofulllem3  27843
  Copyright terms: Public domain W3C validator