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Theorem elrelimasn 6103
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 6102 . . 3 (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥𝐴𝑅𝑥})
21eleq2d 2826 . 2 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥𝐴𝑅𝑥}))
3 brrelex2 5738 . . . 4 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
43ex 412 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐵 ∈ V))
5 breq2 5146 . . . 4 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
65elab3g 3684 . . 3 ((𝐴𝑅𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
74, 6syl 17 . 2 (Rel 𝑅 → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
82, 7bitrd 279 1 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2107  {cab 2713  Vcvv 3479  {csn 4625   class class class wbr 5142  cima 5687  Rel wrel 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  eliniseg2  6123  dprd2dlem2  20061  dprd2dlem1  20062  dprd2da  20063  dprd2d2  20065  dpjfval  20076  ustuqtop4  24254  utop2nei  24260  utop3cls  24261  ucncn  24295  extdgval  33706  cnambfre  37676  frege133d  43783  nzin  44342
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