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Theorem elpreima 5656
Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
elpreima  |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  <->  ( B  e.  A  /\  ( F `  B )  e.  C ) ) )

Proof of Theorem elpreima
StepHypRef Expression
1 cnvimass 5009 . . . . 5  |-  ( `' F " C ) 
C_  dom  F
21sseli 3166 . . . 4  |-  ( B  e.  ( `' F " C )  ->  B  e.  dom  F )
3 fndm 5334 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq2d 2259 . . . 4  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
52, 4imbitrid 154 . . 3  |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  ->  B  e.  A )
)
6 fnfun 5332 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
7 fvimacnvi 5651 . . . . 5  |-  ( ( Fun  F  /\  B  e.  ( `' F " C ) )  -> 
( F `  B
)  e.  C )
86, 7sylan 283 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  ( `' F " C ) )  ->  ( F `  B )  e.  C
)
98ex 115 . . 3  |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  -> 
( F `  B
)  e.  C ) )
105, 9jcad 307 . 2  |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  -> 
( B  e.  A  /\  ( F `  B
)  e.  C ) ) )
11 fvimacnv 5652 . . . . 5  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( ( F `  B )  e.  C  <->  B  e.  ( `' F " C ) ) )
1211funfni 5335 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  e.  C  <->  B  e.  ( `' F " C ) ) )
1312biimpd 144 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  e.  C  ->  B  e.  ( `' F " C ) ) )
1413expimpd 363 . 2  |-  ( F  Fn  A  ->  (
( B  e.  A  /\  ( F `  B
)  e.  C )  ->  B  e.  ( `' F " C ) ) )
1510, 14impbid 129 1  |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  <->  ( B  e.  A  /\  ( F `  B )  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2160   `'ccnv 4643   dom cdm 4644   "cima 4647   Fun wfun 5229    Fn wfn 5230   ` cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  fniniseg  5657  fncnvima2  5658  rexsupp  5661  unpreima  5662  respreima  5665  fisumss  11432  fprodssdc  11630  tanvalap  11748  1arith  12399  ghmpreima  13205  ghmnsgpreima  13208  kerf1ghm  13213  psrbaglesuppg  13950  cncnpi  14188  cncnp  14190  cnpdis  14202  tx1cn  14229  tx2cn  14230  txcnmpt  14233  txdis1cn  14238  xmeterval  14395  cnbl0  14494  cnblcld  14495
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