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Theorem funfvima 5526
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 4734 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21elin2 3188 . . . . . 6  |-  ( B  e.  dom  ( F  |`  A )  <->  ( B  e.  A  /\  B  e. 
dom  F ) )
3 funres 5055 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
4 fvelrn 5430 . . . . . . . . 9  |-  ( ( Fun  ( F  |`  A )  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
53, 4sylan 277 . . . . . . . 8  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
6 fvres 5329 . . . . . . . . . 10  |-  ( B  e.  A  ->  (
( F  |`  A ) `
 B )  =  ( F `  B
) )
76eleq1d 2156 . . . . . . . . 9  |-  ( B  e.  A  ->  (
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) ) )
8 df-ima 4451 . . . . . . . . . 10  |-  ( F
" A )  =  ran  ( F  |`  A )
98eleq2i 2154 . . . . . . . . 9  |-  ( ( F `  B )  e.  ( F " A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) )
107, 9syl6rbbr 197 . . . . . . . 8  |-  ( B  e.  A  ->  (
( F `  B
)  e.  ( F
" A )  <->  ( ( F  |`  A ) `  B )  e.  ran  ( F  |`  A ) ) )
115, 10syl5ibrcom 155 . . . . . . 7  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
1211ex 113 . . . . . 6  |-  ( Fun 
F  ->  ( B  e.  dom  ( F  |`  A )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
132, 12syl5bir 151 . . . . 5  |-  ( Fun 
F  ->  ( ( B  e.  A  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1413expd 254 . . . 4  |-  ( Fun 
F  ->  ( B  e.  A  ->  ( B  e.  dom  F  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) ) ) )
1514com12 30 . . 3  |-  ( B  e.  A  ->  ( Fun  F  ->  ( B  e.  dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) ) )
1615impd 251 . 2  |-  ( B  e.  A  ->  (
( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1716pm2.43b 51 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 102    e. wcel 1438   dom cdm 4438   ran crn 4439    |` cres 4440   "cima 4441   Fun wfun 5009   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  funfvima2  5527  fiintim  6637  djuun  6758
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