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Theorem elimasng1 6107
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5149 and to prove elimasn1 6108 from it. (Revised by BJ, 16-Oct-2024.)
Assertion
Ref Expression
elimasng1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))

Proof of Theorem elimasng1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . 2 ((𝐵𝑉𝐶𝑊) → 𝐶𝑊)
2 imasng 6104 . . 3 (𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
32adantr 480 . 2 ((𝐵𝑉𝐶𝑊) → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
4 simpr 484 . . 3 (((𝐵𝑉𝐶𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
54breq2d 5160 . 2 (((𝐵𝑉𝐶𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥𝐵𝐴𝐶))
61, 3, 5elabd2 3670 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  {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:  elimasn1  6108  elimasng  6109  elimasni  6112  elinisegg  6114
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