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Theorem imasng 5944
Description: The image of a singleton. (Contributed by NM, 8-May-2005.)
Assertion
Ref Expression
imasng (𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem imasng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3510 . 2 (𝐴𝐵𝐴 ∈ V)
2 dfima2 5924 . . 3 (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3 breq1 5060 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
43rexsng 4606 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦𝐴𝑅𝑦))
54abbidv 2882 . . 3 (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦𝐴𝑅𝑦})
62, 5syl5eq 2865 . 2 (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
71, 6syl 17 1 (𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492  {csn 4557   class class class wbr 5057  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  relimasn  5945  elimasn  5947  args  5950  suppvalbr  7823  dfec2  8281  dfac3  9535  shftfib  14419  areacirclem5  34867  dfcoll2  40465  dfatsnafv2  43328
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