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Theorem relimasn 6105
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 4722 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
2 imaeq2 6076 . . . . . . 7 ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
31, 2sylbi 217 . . . . . 6 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4 ima0 6097 . . . . . 6 (𝑅 “ ∅) = ∅
53, 4eqtrdi 2791 . . . . 5 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
65adantl 481 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7 brrelex1 5742 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝑦) → 𝐴 ∈ V)
87stoic1a 1769 . . . . . 6 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
98nexdv 1934 . . . . 5 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10 abn0 4391 . . . . . 6 ({𝑦𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
1110necon1bbii 2988 . . . . 5 (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦𝐴𝑅𝑦} = ∅)
129, 11sylib 218 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦𝐴𝑅𝑦} = ∅)
136, 12eqtr4d 2778 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1413ex 412 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦}))
15 imasng 6104 . 2 (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1614, 15pm2.61d2 181 1 (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  Vcvv 3478  c0 4339  {csn 4631   class class class wbr 5148  cima 5692  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  elrelimasn  6106  predep  6353  funfv2  6997  mapsnd  8925  nznngen  44312  nzss  44313  hashnzfz  44316
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