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Theorem funfvima 5746
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 4927 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
21elin2 3323 . . . . . 6  |-  ( B  e.  dom  ( F  |`  A )  <->  ( B  e.  A  /\  B  e. 
dom  F ) )
3 funres 5256 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
4 fvelrn 5646 . . . . . . . . 9  |-  ( ( Fun  ( F  |`  A )  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
53, 4sylan 283 . . . . . . . 8  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A ) )
6 df-ima 4638 . . . . . . . . . 10  |-  ( F
" A )  =  ran  ( F  |`  A )
76eleq2i 2244 . . . . . . . . 9  |-  ( ( F `  B )  e.  ( F " A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) )
8 fvres 5538 . . . . . . . . . 10  |-  ( B  e.  A  ->  (
( F  |`  A ) `
 B )  =  ( F `  B
) )
98eleq1d 2246 . . . . . . . . 9  |-  ( B  e.  A  ->  (
( ( F  |`  A ) `  B
)  e.  ran  ( F  |`  A )  <->  ( F `  B )  e.  ran  ( F  |`  A ) ) )
107, 9bitr4id 199 . . . . . . . 8  |-  ( B  e.  A  ->  (
( F `  B
)  e.  ( F
" A )  <->  ( ( F  |`  A ) `  B )  e.  ran  ( F  |`  A ) ) )
115, 10syl5ibrcom 157 . . . . . . 7  |-  ( ( Fun  F  /\  B  e.  dom  ( F  |`  A ) )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
1211ex 115 . . . . . 6  |-  ( Fun 
F  ->  ( B  e.  dom  ( F  |`  A )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
132, 12biimtrrid 153 . . . . 5  |-  ( Fun 
F  ->  ( ( B  e.  A  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1413expd 258 . . . 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 254 . 2  |-  ( B  e.  A  ->  (
( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
1716pm2.43b 52 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 104    e. wcel 2148   dom cdm 4625   ran crn 4626    |` cres 4627   "cima 4628   Fun wfun 5209   ` cfv 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223
This theorem is referenced by:  funfvima2  5747  fiintim  6925  caseinl  7087  caseinr  7088  ctssdccl  7107  suplocexprlemdisj  7716  suplocexprlemub  7719  ennnfonelemex  12407  ctinfomlemom  12420  txcnp  13642
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