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Theorem fnimadisj 5355
Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
fnimadisj ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)

Proof of Theorem fnimadisj
StepHypRef Expression
1 fndm 5334 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21ineq1d 3350 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐶) = (𝐴𝐶))
32eqeq1d 2198 . . 3 (𝐹 Fn 𝐴 → ((dom 𝐹𝐶) = ∅ ↔ (𝐴𝐶) = ∅))
43biimpar 297 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
5 imadisj 5008 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
64, 5sylibr 134 1 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  cin 3143  c0 3437  dom cdm 4644  cima 4647   Fn wfn 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fn 5238
This theorem is referenced by: (None)
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