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Theorem imasng 4633
 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 2560 . 2 (A BA V)
2 dfima2 4613 . . 3 (𝑅 “ {A}) = {yx {A}x𝑅y}
3 breq1 3758 . . . . 5 (x = A → (x𝑅yA𝑅y))
43rexsng 3403 . . . 4 (A V → (x {A}x𝑅yA𝑅y))
54abbidv 2152 . . 3 (A V → {yx {A}x𝑅y} = {yA𝑅y})
62, 5syl5eq 2081 . 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 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  {csn 3367   class class class wbr 3755   “ cima 4291 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by:  elreimasng  4634  elimasn  4635  args  4637  fnsnfv  5175  funfvdm2  5180  dfec2  6045
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