Theorem elinisegg 6051

Description: Membership in the inverse image of a singleton. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Put in closed form and shorten proof. (Revised by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
elinisegg	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Proof of Theorem elinisegg

Step	Hyp	Ref	Expression
1		elimasng1 6045	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶))
2		brcnvg 5823	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
3	1, 2	bitrd 281	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  {csn 4557   class class class wbr 5074  ^◡ccnv 5619   “ cima 5623

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633

This theorem is referenced by:  eliniseg  6052  elpredgg  6268