Theorem epin 6085

Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) Put in closed form. (Revised by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
epin	⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)

Proof of Theorem epin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3470	. . . 4 ⊢ 𝑥 ∈ V
2	1	eliniseg 6084	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴))
3		epelg 5572	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴))
4	2, 3	bitrd 279	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴))
5	4	eqrdv 2722	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {csn 4621   class class class wbr 5139   E cep 5570  ^◡ccnv 5666   “ cima 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680

This theorem is referenced by:  epini  6086