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Theorem fimacnvdisj 5419
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 4700 . . . 4  |-  ran  F  =  dom  `' F
2 frn 5395 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
32adantr 451 . . . 4  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ran  F 
C_  B )
41, 3syl5eqssr 3223 . . 3  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  dom  `' F  C_  B )
5 ssdisj 3504 . . 3  |-  ( ( dom  `' F  C_  B  /\  ( B  i^i  C )  =  (/) )  -> 
( dom  `' F  i^i  C )  =  (/) )
64, 5sylancom 648 . 2  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( dom  `' F  i^i  C )  =  (/) )
7 imadisj 5032 . 2  |-  ( ( `' F " C )  =  (/)  <->  ( dom  `' F  i^i  C )  =  (/) )
86, 7sylibr 203 1  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    i^i cin 3151    C_ wss 3152   (/)c0 3455   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251
This theorem is referenced by:  cantnf0  7376  vdwmc2  13026  gsumval3a  15189  psrbag0  16235  mbfconstlem  18984  itg1val2  19039
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-f 5259
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