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Theorem funfvima 5443
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 4680 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21elin2 3170 . . . . . 6  |-  ( B  e.  dom  ( F  |`  A )  <->  ( B  e.  A  /\  B  e. 
dom  F ) )
3 funres 4991 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
4 fvelrn 5351 . . . . . . . . 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 5251 . . . . . . . . . 10  |-  ( B  e.  A  ->  (
( F  |`  A ) `
 B )  =  ( F `  B
) )
76eleq1d 2151 . . . . . . . . 9  |-  ( B  e.  A  ->  (
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) ) )
8 df-ima 4404 . . . . . . . . . 10  |-  ( F
" A )  =  ran  ( F  |`  A )
98eleq2i 2149 . . . . . . . . 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 1434   dom cdm 4391   ran crn 4392    |` cres 4393   "cima 4394   Fun wfun 4946   ` cfv 4952
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  funfvima2  5444  djuun  6568
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