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Theorem fnimadisj 6634
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 6606 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21ineq1d 4172 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐶) = (𝐴𝐶))
32eqeq1d 2735 . . 3 (𝐹 Fn 𝐴 → ((dom 𝐹𝐶) = ∅ ↔ (𝐴𝐶) = ∅))
43biimpar 479 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
5 imadisj 6033 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
64, 5sylibr 233 1 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cin 3910  c0 4283  dom cdm 5634  cima 5637   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fn 6500
This theorem is referenced by:  poimirlem15  36139  aacllem  47334
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