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Theorem elimasng1 6042
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5110 and to prove elimasn1 6043 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 486 . 2 ((𝐵𝑉𝐶𝑊) → 𝐶𝑊)
2 imasng 6039 . . 3 (𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
32adantr 482 . 2 ((𝐵𝑉𝐶𝑊) → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
4 simpr 486 . . 3 (((𝐵𝑉𝐶𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
54breq2d 5121 . 2 (((𝐵𝑉𝐶𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥𝐵𝐴𝐶))
61, 3, 5elabd2 3626 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  {csn 4590   class class class wbr 5109  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  elimasn1  6043  elimasng  6044  elimasni  6047  elinisegg  6049
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