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Theorem imasng 4904
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 2697 . 2 (𝐴𝐵𝐴 ∈ V)
2 dfima2 4883 . . 3 (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦}
3 breq1 3932 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
43rexsng 3565 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦𝐴𝑅𝑦))
54abbidv 2257 . . 3 (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦𝐴𝑅𝑦})
62, 5syl5eq 2184 . 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 1331  wcel 1480  {cab 2125  wrex 2417  Vcvv 2686  {csn 3527   class class class wbr 3929  cima 4542
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552
This theorem is referenced by:  elreimasng  4905  elimasn  4906  args  4908  fnsnfv  5480  funfvdm2  5485  dfec2  6432  mapsn  6584  shftfibg  10599  shftfib  10602
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