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Theorem imasng 6039
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 3465 . 2 (𝐴𝐵𝐴 ∈ V)
2 dfima2 6019 . . 3 (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3 breq1 5112 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
43rexsng 4639 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦𝐴𝑅𝑦))
54abbidv 2802 . . 3 (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦𝐴𝑅𝑦})
62, 5eqtrid 2785 . 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 1542  wcel 2107  {cab 2710  wrex 3070  Vcvv 3447  {csn 4590   class class class wbr 5109  cima 5640
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-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  relimasn  6040  elimasng1  6042  args  6048  fnsnfv  6924  suppvalbr  8100  dfec2  8657  dfac3  10065  shftfib  14966  areacirclem5  36220  dfcoll2  42624  dfatsnafv2  45574
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