Theorem elimaksn 4284

Description: Membership in a Kuratowski image of a singleton. (Contributed by SF, 4-Feb-2015.)

Hypotheses

Ref	Expression
elimaksn.1	|- B e. _V
elimaksn.2	|- C e. _V

Assertion

Ref	Expression
elimaksn	|- (C e. (A"k{B}) <-> <<B, C>> e. A)

Proof of Theorem elimaksn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimaksn.2	. . 3 |- C e. _V
2	1	elimak 4260	. 2 |- (C e. (A"k{B}) <-> E.x e. {B}<<x, C>> e. A)
3		elimaksn.1	. . 3 |- B e. _V
4		opkeq1 4060	. . . 4 |- (x = B -> <<x, C>> = <<B, C>>)
5	4	eleq1d 2419	. . 3 |- (x = B -> (<<x, C>> e. A <-> <<B, C>> e. A))
6	3, 5	rexsn 3769	. 2 |- (E.x e. {B}<<x, C>> e. A <-> <<B, C>> e. A)
7	2, 6	bitri 240	1 |- (C e. (A"k{B}) <-> <<B, C>> e. A)

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860  {csn 3738  <<copk 4058  "_kcimak 4180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059  df-imak 4190

This theorem is referenced by:  preaddccan2lem1  4455