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Theorem relimasn 6102
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 4716 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
2 imaeq2 6073 . . . . . . 7 ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
31, 2sylbi 217 . . . . . 6 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4 ima0 6094 . . . . . 6 (𝑅 “ ∅) = ∅
53, 4eqtrdi 2792 . . . . 5 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
65adantl 481 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7 brrelex1 5737 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝑦) → 𝐴 ∈ V)
87stoic1a 1771 . . . . . 6 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
98nexdv 1935 . . . . 5 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10 abn0 4384 . . . . . 6 ({𝑦𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
1110necon1bbii 2989 . . . . 5 (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦𝐴𝑅𝑦} = ∅)
129, 11sylib 218 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦𝐴𝑅𝑦} = ∅)
136, 12eqtr4d 2779 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1413ex 412 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦}))
15 imasng 6101 . 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 1539  wex 1778  wcel 2107  {cab 2713  Vcvv 3479  c0 4332  {csn 4625   class class class wbr 5142  cima 5687  Rel wrel 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  elrelimasn  6103  predep  6350  funfv2  6996  mapsnd  8927  nznngen  44340  nzss  44341  hashnzfz  44344
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