Theorem elimasn1 5944

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 5044 and shorten. (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 5943	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)

Syntax hints:   ↔ wb 209   ∈ wcel 2110  Vcvv 3401  {csn 4531   class class class wbr 5043   “ cima 5543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-br 5044  df-opab 5106  df-xp 5546  df-cnv 5548  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553

This theorem is referenced by: (None)