Theorem epini 6046

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 6045	. 2 ⊢ (𝐴 ∈ V → (◡ E “ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3443  {csn 4584   E cep 5534  ^◡ccnv 5630   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-eprel 5535  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  infxpenlem  9907  fz1isolem  14314