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Theorem elimasn1 6108
Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5149 and shorten proof. (Revised by BJ, 16-Oct-2024.)
Hypotheses
Ref Expression
elimasn1.1 𝐵 ∈ V
elimasn1.2 𝐶 ∈ V
Assertion
Ref Expression
elimasn1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)

Proof of Theorem elimasn1
StepHypRef Expression
1 elimasn1.1 . 2 𝐵 ∈ V
2 elimasn1.2 . 2 𝐶 ∈ V
3 elimasng1 6107 . 2 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
41, 2, 3mp2an 692 1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  Vcvv 3478  {csn 4631   class class class wbr 5148  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by: (None)
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