Theorem epini 6067

Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Hypothesis

Ref	Expression
epini.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
epini	⊢ (◡ E “ {𝐴}) = 𝐴

Proof of Theorem epini

Step	Hyp	Ref	Expression
1		epini.1	. 2 ⊢ 𝐴 ∈ V
2		epin 6066	. 2 ⊢ (𝐴 ∈ V → (◡ E “ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {csn 4589   E cep 5537  ^◡ccnv 5637   “ cima 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-eprel 5538  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  infxpenlem  9966  fz1isolem  14426