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Theorem fnimaeq0 5365
Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 27148. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Assertion
Ref Expression
fnimaeq0  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )

Proof of Theorem fnimaeq0
StepHypRef Expression
1 imadisj 5032 . 2  |-  ( ( F " B )  =  (/)  <->  ( dom  F  i^i  B )  =  (/) )
2 incom 3361 . . . 4  |-  ( dom 
F  i^i  B )  =  ( B  i^i  dom 
F )
3 fndm 5343 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
43sseq2d 3206 . . . . . 6  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
54biimpar 471 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
6 df-ss 3166 . . . . 5  |-  ( B 
C_  dom  F  <->  ( B  i^i  dom  F )  =  B )
75, 6sylib 188 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( B  i^i  dom  F )  =  B )
82, 7syl5eq 2327 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( dom  F  i^i  B )  =  B )
98eqeq1d 2291 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( dom  F  i^i  B )  =  (/)  <->  B  =  (/) ) )
101, 9syl5bb 248 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    i^i cin 3151    C_ wss 3152   (/)c0 3455   dom cdm 4689   "cima 4692    Fn wfn 5250
This theorem is referenced by:  ipodrsima  14268  mdegldg  19452  ig1peu  19557  ig1pdvds  19562  kelac1  27161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fn 5258
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