Theorem elimasng1 6105

Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5144 and to prove elimasn1 6106 from it. (Revised by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
elimasng1	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))

Proof of Theorem elimasng1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ 𝑊)
2		imasng 6102	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
3	2	adantr 480	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
4		simpr 484	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
5	4	breq2d 5155	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶))
6	1, 3, 5	elabd2 3670	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  {csn 4626   class class class wbr 5143   “ cima 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  elimasn1  6106  elimasng  6107  elimasni  6109  elinisegg  6111