Theorem epini 5954

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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		epini.1	. . . 4 ⊢ 𝐴 ∈ V
2		vex 3498	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eliniseg 5953	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴))
4	1, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴)
5	1	epeli 5463	. . 3 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
6	4, 5	bitri 277	. 2 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴)
7	6	eqriv 2818	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   ↔ wb 208   = wceq 1533   ∈ wcel 2110  Vcvv 3495  {csn 4561   class class class wbr 5059   E cep 5459  ^◡ccnv 5549   “ cima 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-eprel 5460  df-xp 5556  df-cnv 5558  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563

This theorem is referenced by:  infxpenlem  9433  fz1isolem  13813