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Theorem funfvima 5965
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 5159 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21elin2 3523 . . . . . 6  |-  ( B  e.  dom  ( F  |`  A )  <->  ( B  e.  A  /\  B  e. 
dom  F ) )
3 funres 5484 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
4 fvelrn 5858 . . . . . . . . 9  |-  ( ( Fun  ( F  |`  A )  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
53, 4sylan 458 . . . . . . . 8  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
6 fvres 5737 . . . . . . . . . 10  |-  ( B  e.  A  ->  (
( F  |`  A ) `
 B )  =  ( F `  B
) )
76eleq1d 2501 . . . . . . . . 9  |-  ( B  e.  A  ->  (
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) ) )
8 df-ima 4883 . . . . . . . . . 10  |-  ( F
" A )  =  ran  ( F  |`  A )
98eleq2i 2499 . . . . . . . . 9  |-  ( ( F `  B )  e.  ( F " A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) )
107, 9syl6rbbr 256 . . . . . . . 8  |-  ( B  e.  A  ->  (
( F `  B
)  e.  ( F
" A )  <->  ( ( F  |`  A ) `  B )  e.  ran  ( F  |`  A ) ) )
115, 10syl5ibrcom 214 . . . . . . 7  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
1211ex 424 . . . . . 6  |-  ( Fun 
F  ->  ( B  e.  dom  ( F  |`  A )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
132, 12syl5bir 210 . . . . 5  |-  ( Fun 
F  ->  ( ( B  e.  A  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1413exp3a 426 . . . 4  |-  ( Fun 
F  ->  ( B  e.  A  ->  ( B  e.  dom  F  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) ) ) )
1514com12 29 . . 3  |-  ( B  e.  A  ->  ( Fun  F  ->  ( B  e.  dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) ) )
1615imp3a 421 . 2  |-  ( B  e.  A  ->  (
( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1716pm2.43b 48 1  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440   ` cfv 5446
This theorem is referenced by:  funfvima2  5966  tz7.48-2  6691  tz9.12lem3  7707  txcnp  17644  c1liplem1  19872  htthlem  22412  elovimad  24043  tpr2rico  24302  brsiga  24529  erdszelem8  24876  nobndlem2  25640  nofulllem3  25651  lindff1  27258
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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