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Theorem fnimaeq0 5507
Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 26818. (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 5164 . 2  |-  ( ( F " B )  =  (/)  <->  ( dom  F  i^i  B )  =  (/) )
2 incom 3477 . . . 4  |-  ( dom 
F  i^i  B )  =  ( B  i^i  dom 
F )
3 fndm 5485 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
43sseq2d 3320 . . . . . 6  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
54biimpar 472 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
6 df-ss 3278 . . . . 5  |-  ( B 
C_  dom  F  <->  ( B  i^i  dom  F )  =  B )
75, 6sylib 189 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( B  i^i  dom  F )  =  B )
82, 7syl5eq 2432 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( dom  F  i^i  B )  =  B )
98eqeq1d 2396 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( dom  F  i^i  B )  =  (/)  <->  B  =  (/) ) )
101, 9syl5bb 249 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    i^i cin 3263    C_ wss 3264   (/)c0 3572   dom cdm 4819   "cima 4822    Fn wfn 5390
This theorem is referenced by:  ipodrsima  14519  mdegldg  19857  ig1peu  19962  ig1pdvds  19967  kelac1  26831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fn 5398
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