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Theorem ndmima 5944
 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 5926 . 2 ((𝐵 “ {𝐴}) = ∅ ↔ (dom 𝐵 ∩ {𝐴}) = ∅)
2 disjsn 4621 . 2 ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵)
31, 2sylbbr 239 1 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2114   ∩ cin 3907  ∅c0 4265  {csn 4539  dom cdm 5532   “ cima 5535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545 This theorem is referenced by:  funfv  6732  dffv2  6738  fpwwe2lem13  10053  bj-funun  34628
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