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Theorem ndmima 6059
Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.)
Assertion
Ref Expression
ndmima 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)

Proof of Theorem ndmima
StepHypRef Expression
1 imadisj 6036 . 2 ((𝐵 “ {𝐴}) = ∅ ↔ (dom 𝐵 ∩ {𝐴}) = ∅)
2 disjsn 4676 . 2 ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵)
31, 2sylbbr 235 1 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  cin 3913  c0 4286  {csn 4590  dom cdm 5637  cima 5640
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 5260  ax-nul 5267  ax-pr 5388
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 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  funfv  6932  dffv2  6940  imafi  9125  fpwwe2lem12  10586  bj-funun  35773
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