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Theorem fimacnvdisj 5195
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 4449 . . . 4  |-  ran  F  =  dom  `' F
2 frn 5169 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
32adantr 270 . . . 4  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ran  F 
C_  B )
41, 3syl5eqssr 3071 . . 3  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  dom  `' F  C_  B )
5 ssdisj 3339 . . 3  |-  ( ( dom  `' F  C_  B  /\  ( B  i^i  C )  =  (/) )  -> 
( dom  `' F  i^i  C )  =  (/) )
64, 5sylancom 411 . 2  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( dom  `' F  i^i  C )  =  (/) )
7 imadisj 4794 . 2  |-  ( ( `' F " C )  =  (/)  <->  ( dom  `' F  i^i  C )  =  (/) )
86, 7sylibr 132 1  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    i^i cin 2998    C_ wss 2999   (/)c0 3286   `'ccnv 4437   dom cdm 4438   ran crn 4439   "cima 4441   -->wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-f 5019
This theorem is referenced by: (None)
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