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Theorem imasng 4874
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 2671 . 2 (𝐴𝐵𝐴 ∈ V)
2 dfima2 4853 . . 3 (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3 breq1 3902 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
43rexsng 3535 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦𝐴𝑅𝑦))
54abbidv 2235 . . 3 (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦𝐴𝑅𝑦})
62, 5syl5eq 2162 . 2 (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
71, 6syl 14 1 (𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  {cab 2103  wrex 2394  Vcvv 2660  {csn 3497   class class class wbr 3899  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  elreimasng  4875  elimasn  4876  args  4878  fnsnfv  5448  funfvdm2  5453  dfec2  6400  mapsn  6552  shftfibg  10560  shftfib  10563
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