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Theorem fnimaeq0 5707
Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 35980. (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 5193 . 2  |-  ( ( F " B )  =  (/)  <->  ( dom  F  i^i  B )  =  (/) )
2 incom 3616 . . . 4  |-  ( dom 
F  i^i  B )  =  ( B  i^i  dom 
F )
3 fndm 5685 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
43sseq2d 3446 . . . . . 6  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
54biimpar 493 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
6 df-ss 3404 . . . . 5  |-  ( B 
C_  dom  F  <->  ( B  i^i  dom  F )  =  B )
75, 6sylib 201 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( B  i^i  dom  F )  =  B )
82, 7syl5eq 2517 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( dom  F  i^i  B )  =  B )
98eqeq1d 2473 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( dom  F  i^i  B )  =  (/)  <->  B  =  (/) ) )
101, 9syl5bb 265 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    i^i cin 3389    C_ wss 3390   (/)c0 3722   dom cdm 4839   "cima 4842    Fn wfn 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fn 5592
This theorem is referenced by:  ipodrsima  16489  mdegldg  23094  ig1peu  23201  ig1peuOLD  23202  ig1pdvds  23207  ig1pdvdsOLD  23213  kelac1  35992
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