Theorem epini 4975

Description: Any set is equal to its preimage under the converse epsilon 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 2729	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eliniseg 4974	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴))
4	1, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 E 𝐴)
5	1	epelc 4269	. . 3 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
6	4, 5	bitri 183	. 2 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴)
7	6	eqriv 2162	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   ↔ wb 104   = wceq 1343   ∈ wcel 2136  Vcvv 2726  {csn 3576   class class class wbr 3982   E cep 4265  ^◡ccnv 4603   “ cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by: (None)