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Theorem fimacnvdisj 5778
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 4865 . . . 4  |-  ran  F  =  dom  `' F
2 frn 5752 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
32adantr 466 . . . 4  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ran  F 
C_  B )
41, 3syl5eqssr 3515 . . 3  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  dom  `' F  C_  B )
5 ssdisj 3848 . . 3  |-  ( ( dom  `' F  C_  B  /\  ( B  i^i  C )  =  (/) )  -> 
( dom  `' F  i^i  C )  =  (/) )
64, 5sylancom 671 . 2  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( dom  `' F  i^i  C )  =  (/) )
7 imadisj 5207 . 2  |-  ( ( `' F " C )  =  (/)  <->  ( dom  `' F  i^i  C )  =  (/) )
86, 7sylibr 215 1  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    i^i cin 3441    C_ wss 3442   (/)c0 3767   `'ccnv 4853   dom cdm 4854   ran crn 4855   "cima 4857   -->wf 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-f 5605
This theorem is referenced by:  vdwmc2  14892  gsumval3a  17472  psrbag0  18652  mbfconstlem  22462  itg1val2  22519  ofpreima2  28109
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