Theorem epin 5992

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 3426	. . . 4 ⊢ 𝑥 ∈ V
2	1	eliniseg 5991	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴))
3		epelg 5487	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴))
4	2, 3	bitrd 278	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴))
5	4	eqrdv 2736	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {csn 4558   class class class wbr 5070   E cep 5485  ^◡ccnv 5579   “ cima 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-eprel 5486  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by:  epini  5993