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Theorem relimasn 6084
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 6056 . . . . . . 7 ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
31, 2sylbi 216 . . . . . 6 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4 ima0 6077 . . . . . 6 (𝑅 “ ∅) = ∅
53, 4eqtrdi 2789 . . . . 5 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
65adantl 483 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7 brrelex1 5730 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝑦) → 𝐴 ∈ V)
87stoic1a 1775 . . . . . 6 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
98nexdv 1940 . . . . 5 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10 abn0 4381 . . . . . 6 ({𝑦𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
1110necon1bbii 2991 . . . . 5 (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦𝐴𝑅𝑦} = ∅)
129, 11sylib 217 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦𝐴𝑅𝑦} = ∅)
136, 12eqtr4d 2776 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1413ex 414 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦}))
15 imasng 6083 . 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 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  Vcvv 3475  c0 4323  {csn 4629   class class class wbr 5149  cima 5680  Rel wrel 5682
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 5300  ax-nul 5307  ax-pr 5428
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-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  elrelimasn  6085  predep  6332  fnsnfvOLD  6972  funfv2  6980  mapsnd  8880  nznngen  43075  nzss  43076  hashnzfz  43079
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