Theorem epini 6095

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 6094	. 2 ⊢ (𝐴 ∈ V → (◡ E “ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4628   E cep 5579  ^◡ccnv 5675   “ cima 5679

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-eprel 5580  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by:  infxpenlem  10007  fz1isolem  14421