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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 2541 . 2 (A BA V)
2 dfima2 4595 . . 3 (𝑅 “ {A}) = {yx {A}x𝑅y}
3 breq1 3739 . . . . 5 (x = A → (x𝑅yA𝑅y))
43rexsng 3384 . . . 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 1228   wcel 1374  {cab 2008  wrex 2283  Vcvv 2533  {csn 3348   class class class wbr 3736  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  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 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-br 3737  df-opab 3791  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  6018
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