Theorem epini 5156

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 2824	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eliniseg 5155	. . . 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 4434	. . 3 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
6	4, 5	bitri 184	. 2 ⊢ (𝑥 ∈ (◡ E “ {𝐴}) ↔ 𝑥 ∈ 𝐴)
7	6	eqriv 2235	1 ⊢ (◡ E “ {𝐴}) = 𝐴

Syntax hints:   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3708   class class class wbr 4128   E cep 4430  ^◡ccnv 4771   “ cima 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-eprel 4432  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785

This theorem is referenced by: (None)