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Theorem elreimasng 4634
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 4633 . . 3 (A 𝑉 → (𝑅 “ {A}) = {xA𝑅x})
21eleq2d 2104 . 2 (A 𝑉 → (B (𝑅 “ {A}) ↔ B {xA𝑅x}))
3 brrelex2 4326 . . . 4 ((Rel 𝑅 A𝑅B) → B V)
43ex 108 . . 3 (Rel 𝑅 → (A𝑅BB V))
5 breq2 3759 . . . 4 (x = B → (A𝑅xA𝑅B))
65elab3g 2687 . . 3 ((A𝑅BB V) → (B {xA𝑅x} ↔ A𝑅B))
74, 6syl 14 . 2 (Rel 𝑅 → (B {xA𝑅x} ↔ A𝑅B))
82, 7sylan9bbr 436 1 ((Rel 𝑅 A 𝑉) → (B (𝑅 “ {A}) ↔ A𝑅B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  {cab 2023  Vcvv 2551  {csn 3367   class class class wbr 3755  cima 4291  Rel wrel 4293
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-bnd 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-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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