Theorem epini 6096

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 6095	. 2 ⊢ (𝐴 ∈ V → (◡ E “ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (◡ E “ {𝐴}) = 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4584   E cep 5550  ^◡ccnv 5650   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-eprel 5551  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  infxpenlem  10099  fz1isolem  14613  j0ltalephlem  35950