Theorem elimasn1 6099

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 5156 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

Step	Hyp	Ref	Expression
1		elimasn1.1	. 2 ⊢ 𝐵 ∈ V
2		elimasn1.2	. 2 ⊢ 𝐶 ∈ V
3		elimasng1 6098	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)

Syntax hints:   ↔ wb 205   ∈ wcel 2099  Vcvv 3462  {csn 4633   class class class wbr 5155   “ cima 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5156  df-opab 5218  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697

This theorem is referenced by: (None)