Theorem ndmima 6089

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

Step	Hyp	Ref	Expression
1		imadisj 6066	. 2 ⊢ ((𝐵 “ {𝐴}) = ∅ ↔ (dom 𝐵 ∩ {𝐴}) = ∅)
2		disjsn 4669	. 2 ⊢ ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵)
3	1, 2	sylbbr 238	1 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1559   ∈ wcel 2141   ∩ cin 3903  ∅c0 4285  {csn 4581  dom cdm 5645   “ cima 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658

This theorem is referenced by:  funfv  6950  dffv2  6958  imafiOLD  9256  fpwwe2lem12  10597  bj-funun  37708