Theorem epin 6125

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 3492	. . . 4 ⊢ 𝑥 ∈ V
2	1	eliniseg 6124	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴))
3		epelg 5600	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴))
4	2, 3	bitrd 279	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴))
5	4	eqrdv 2738	1 ⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {csn 4648   class class class wbr 5166   E cep 5598  ^◡ccnv 5699   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  epini  6126