ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elreimasng GIF version

Theorem elreimasng 4765
Description: Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
Assertion
Ref Expression
elreimasng ((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elreimasng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imasng 4764 . . 3 (𝐴𝑉 → (𝑅 “ {𝐴}) = {𝑥𝐴𝑅𝑥})
21eleq2d 2154 . 2 (𝐴𝑉 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥𝐴𝑅𝑥}))
3 brrelex2 4449 . . . 4 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
43ex 113 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐵 ∈ V))
5 breq2 3824 . . . 4 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
65elab3g 2757 . . 3 ((𝐴𝑅𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
74, 6syl 14 . 2 (Rel 𝑅 → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
82, 7sylan9bbr 451 1 ((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1436  {cab 2071  Vcvv 2615  {csn 3431   class class class wbr 3820  cima 4414  Rel wrel 4416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4417  df-rel 4418  df-cnv 4419  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator