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Theorem imasng 4615
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 2542 . 2 (A BA V)
2 dfima2 4595 . . 3 (𝑅 “ {A}) = {yx {A}x𝑅y}
3 breq1 3740 . . . . 5 (x = A → (x𝑅yA𝑅y))
43rexsng 3385 . . . 4 (A V → (x {A}x𝑅yA𝑅y))
54abbidv 2138 . . 3 (A V → {yx {A}x𝑅y} = {yA𝑅y})
62, 5syl5eq 2067 . 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 1228   wcel 1375  {cab 2009  wrex 2284  Vcvv 2534  {csn 3349   class class class wbr 3737  cima 4273
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  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 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-br 3738  df-opab 3792  df-xp 4276  df-cnv 4278  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283
This theorem is referenced by:  elreimasng  4616  elimasn  4617  args  4619  fnsnfv  5155  funfvdm2  5160  dfec2  6015
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