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Theorem elreimasng 4618
 Description: Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
Assertion
Ref Expression
elreimasng ((Rel 𝑅 A 𝑉) → (B (𝑅 “ {A}) ↔ A𝑅B))

Proof of Theorem elreimasng
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 imasng 4617 . . 3 (A 𝑉 → (𝑅 “ {A}) = {xA𝑅x})
21eleq2d 2089 . 2 (A 𝑉 → (B (𝑅 “ {A}) ↔ B {xA𝑅x}))
3 brrelex2 4310 . . . 4 ((Rel 𝑅 A𝑅B) → B V)
43ex 108 . . 3 (Rel 𝑅 → (A𝑅BB V))
5 breq2 3742 . . . 4 (x = B → (A𝑅xA𝑅B))
65elab3g 2670 . . 3 ((A𝑅BB V) → (B {xA𝑅x} ↔ A𝑅B))
74, 6syl 14 . 2 (Rel 𝑅 → (B {xA𝑅x} ↔ A𝑅B))
82, 7sylan9bbr 439 1 ((Rel 𝑅 A 𝑉) → (B (𝑅 “ {A}) ↔ A𝑅B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1374  {cab 2008  Vcvv 2535  {csn 3350   class class class wbr 3738   “ cima 4275  Rel wrel 4277 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 3849  ax-pow 3901  ax-pr 3918 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 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285 This theorem is referenced by: (None)
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