Theorem epin 6094

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 3474	. . . 4 ⊢ 𝑥 ∈ V
2	1	eliniseg 6093	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴))
3		epelg 5578	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴))
4	2, 3	bitrd 279	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴))
5	4	eqrdv 2726	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {csn 4625   class class class wbr 5143   E cep 5576  ^◡ccnv 5672   “ cima 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-eprel 5577  df-xp 5679  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686

This theorem is referenced by:  epini  6095