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Theorem fnimadisj 6676
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 6646 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21ineq1d 4206 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐶) = (𝐴𝐶))
32eqeq1d 2728 . . 3 (𝐹 Fn 𝐴 → ((dom 𝐹𝐶) = ∅ ↔ (𝐴𝐶) = ∅))
43biimpar 477 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
5 imadisj 6073 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
64, 5sylibr 233 1 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  cin 3942  c0 4317  dom cdm 5669  cima 5672   Fn wfn 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fn 6540
This theorem is referenced by:  poimirlem15  37016  aacllem  48122
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