Theorem epini 4950

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 2712	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eliniseg 4949	. . . 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 4246	. . 3 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
6	4, 5	bitri 183	. 2 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴)
7	6	eqriv 2151	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 2125  Vcvv 2709  {csn 3556   class class class wbr 3961   E cep 4242  ^◡ccnv 4578   “ cima 4582

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-eprel 4244  df-xp 4585  df-cnv 4587  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592

This theorem is referenced by: (None)