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Theorem fimacnvdisj 5380
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 4620 . . . 4  |-  ran  F  =  dom  `' F
2 frn 5354 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
32adantr 274 . . . 4  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ran  F 
C_  B )
41, 3eqsstrrid 3194 . . 3  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  dom  `' F  C_  B )
5 ssdisj 3470 . . 3  |-  ( ( dom  `' F  C_  B  /\  ( B  i^i  C )  =  (/) )  -> 
( dom  `' F  i^i  C )  =  (/) )
64, 5sylancom 418 . 2  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( dom  `' F  i^i  C )  =  (/) )
7 imadisj 4971 . 2  |-  ( ( `' F " C )  =  (/)  <->  ( dom  `' F  i^i  C )  =  (/) )
86, 7sylibr 133 1  |-  ( ( F : A --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    i^i cin 3120    C_ wss 3121   (/)c0 3414   `'ccnv 4608   dom cdm 4609   ran crn 4610   "cima 4612   -->wf 5192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-f 5200
This theorem is referenced by: (None)
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