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Theorem elrelimasn 5946
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 5945 . . 3 (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑥𝐴𝑅𝑥})
21eleq2d 2896 . 2 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐵 ∈ {𝑥𝐴𝑅𝑥}))
3 brrelex2 5599 . . . 4 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
43ex 415 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐵 ∈ V))
5 breq2 5061 . . . 4 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
65elab3g 3671 . . 3 ((𝐴𝑅𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
74, 6syl 17 . 2 (Rel 𝑅 → (𝐵 ∈ {𝑥𝐴𝑅𝑥} ↔ 𝐴𝑅𝐵))
82, 7bitrd 281 1 (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wcel 2107  {cab 2797  Vcvv 3493  {csn 4559   class class class wbr 5057  cima 5551  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  eliniseg2  5962  dprd2dlem2  19154  dprd2dlem1  19155  dprd2da  19156  dprd2d2  19158  dpjfval  19169  ustuqtop4  22845  utop2nei  22851  utop3cls  22852  ucncn  22886  extdgval  31032  cnambfre  34927  frege133d  40095  nzin  40635
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