Theorem elimasn1 6117

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 5167 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 6116	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
4	1, 2, 3	mp2an 691	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3488  {csn 4648   class class class wbr 5166   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by: (None)