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Theorem relimasn 5928
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 4629 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
2 imaeq2 5901 . . . . . . 7 ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
31, 2sylbi 219 . . . . . 6 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅))
4 ima0 5921 . . . . . 6 (𝑅 “ ∅) = ∅
53, 4syl6eq 2871 . . . . 5 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅)
65adantl 484 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅)
7 brrelex1 5581 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝑦) → 𝐴 ∈ V)
87stoic1a 1773 . . . . . 6 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦)
98nexdv 1937 . . . . 5 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦)
10 abn0 4312 . . . . . 6 ({𝑦𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦)
1110necon1bbii 3055 . . . . 5 (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦𝐴𝑅𝑦} = ∅)
129, 11sylib 220 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦𝐴𝑅𝑦} = ∅)
136, 12eqtr4d 2858 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1413ex 415 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦}))
15 imasng 5927 . 2 (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
1614, 15pm2.61d2 183 1 (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1537  wex 1780  wcel 2114  {cab 2798  Vcvv 3473  c0 4269  {csn 4543   class class class wbr 5042  cima 5534  Rel wrel 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544
This theorem is referenced by:  elrelimasn  5929  predep  6150  fnsnfv  6719  funfv2  6727  mapsnd  8428  fnimasnd  39233  nznngen  40803  nzss  40804  hashnzfz  40807
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