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Theorem relimasn 5407
 Description: The image of a singleton. (Contributed by NM, 20-May-1998.)
Assertion
Ref Expression
relimasn (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅

Proof of Theorem relimasn
StepHypRef Expression
1 snprc 4197 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
2 imaeq2 5381 . . . . . . 7 ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
31, 2sylbi 206 . . . . . 6 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4 ima0 5400 . . . . . 6 (𝑅 “ ∅) = ∅
53, 4syl6eq 2660 . . . . 5 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
65adantl 481 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7 brrelex 5080 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝑦) → 𝐴 ∈ V)
87stoic1a 1688 . . . . . 6 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
98nexdv 1851 . . . . 5 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10 abn0 3908 . . . . . 6 ({𝑦𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
1110necon1bbii 2831 . . . . 5 (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦𝐴𝑅𝑦} = ∅)
129, 11sylib 207 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦𝐴𝑅𝑦} = ∅)
136, 12eqtr4d 2647 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1413ex 449 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦}))
15 imasng 5406 . 2 (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1614, 15pm2.61d2 171 1 (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  Vcvv 3173  ∅c0 3874  {csn 4125   class class class wbr 4583   “ cima 5041  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  elrelimasn  5408  predep  5623  fnsnfv  6168  funfv2  6176  mapsn  7785  nznngen  37537  nzss  37538  hashnzfz  37541  mapsnd  38383
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