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Theorem fimacnvdisj 5769
Description: The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
fimacnvdisj  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )

Proof of Theorem fimacnvdisj
StepHypRef Expression
1 df-rn 5016 . . . 4  |-  ran  F  =  dom  `' F
2 frn 5743 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
32adantr 465 . . . 4  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ran  F 
C_  B )
41, 3syl5eqssr 3554 . . 3  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  dom  `' F  C_  B )
5 ssdisj 3881 . . 3  |-  ( ( dom  `' F  C_  B  /\  ( B  i^i  C )  =  (/) )  -> 
( dom  `' F  i^i  C )  =  (/) )
64, 5sylancom 667 . 2  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( dom  `' F  i^i  C )  =  (/) )
7 imadisj 5362 . 2  |-  ( ( `' F " C )  =  (/)  <->  ( dom  `' F  i^i  C )  =  (/) )
86, 7sylibr 212 1  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    i^i cin 3480    C_ wss 3481   (/)c0 3790   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-f 5598
This theorem is referenced by:  vdwmc2  14373  gsumval3a  16778  gsumval3aOLD  16779  psrbag0  18029  mbfconstlem  21904  itg1val2  21959  ofpreima2  27322
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