MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrelimasn Structured version   Visualization version   GIF version

Theorem elrelimasn 6084
Description: Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
elrelimasn (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elrelimasn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relimasn 6083 . . 3 (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥𝐴𝑅𝑥})
21eleq2d 2819 . 2 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥𝐴𝑅𝑥}))
3 brrelex2 5730 . . . 4 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
43ex 413 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐵 ∈ V))
5 breq2 5152 . . . 4 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
65elab3g 3675 . . 3 ((𝐴𝑅𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
74, 6syl 17 . 2 (Rel 𝑅 → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
82, 7bitrd 278 1 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  {cab 2709  Vcvv 3474  {csn 4628   class class class wbr 5148  cima 5679  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  eliniseg2  6105  dprd2dlem2  19909  dprd2dlem1  19910  dprd2da  19911  dprd2d2  19913  dpjfval  19924  ustuqtop4  23748  utop2nei  23754  utop3cls  23755  ucncn  23789  extdgval  32728  cnambfre  36531  frege133d  42506  nzin  43067
  Copyright terms: Public domain W3C validator