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Theorem imasng 4584
Description: The image of a singleton. (Contributed by NM, 8-May-2005.)
Assertion
Ref Expression
imasng (A B → (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 2541 . 2 (A BA V)
2 dfima2 4564 . . 3 (𝑅 “ {A}) = {yx {A}x𝑅y}
3 breq1 3719 . . . . 5 (x = A → (x𝑅yA𝑅y))
43rexsng 3363 . . . 4 (A V → (x {A}x𝑅yA𝑅y))
54abbidv 2137 . . 3 (A V → {yx {A}x𝑅y} = {yA𝑅y})
62, 5syl5eq 2066 . 2 (A V → (𝑅 “ {A}) = {yA𝑅y})
71, 6syl 14 1 (A B → (𝑅 “ {A}) = {yA𝑅y})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1373   wcel 1375  {cab 2008  wrex 2283  Vcvv 2533  {csn 3327   class class class wbr 3716  cima 4241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-br 3717  df-opab 3771  df-xp 4244  df-cnv 4246  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251
This theorem is referenced by:  elreimasng  4585  elimasn  4586  args  4588  fnsnfv  5124  funfvdm2  5129
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