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Theorem imasng 5446
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 3198 . 2 (𝐴𝐵𝐴 ∈ V)
2 dfima2 5427 . . 3 (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3 breq1 4616 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
43rexsng 4190 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦𝐴𝑅𝑦))
54abbidv 2738 . . 3 (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦𝐴𝑅𝑦})
62, 5syl5eq 2667 . 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 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186  {csn 4148   class class class wbr 4613  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  relimasn  5447  elimasn  5449  args  5452  suppvalbr  7244  dfec2  7690  dfac3  8888  shftfib  13746  areacirclem5  33133
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