Theorem elimaksn 4283

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

Hypotheses

Ref	Expression
elimaksn.1	⊢ B ∈ V
elimaksn.2	⊢ C ∈ V

Assertion

Ref	Expression
elimaksn	⊢ (C ∈ (A “k {B}) ↔ ⟪B, C⟫ ∈ A)

Proof of Theorem elimaksn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimaksn.2	. . 3 ⊢ C ∈ V
2	1	elimak 4259	. 2 ⊢ (C ∈ (A “k {B}) ↔ ∃x ∈ {B}⟪x, C⟫ ∈ A)
3		elimaksn.1	. . 3 ⊢ B ∈ V
4		opkeq1 4059	. . . 4 ⊢ (x = B → ⟪x, C⟫ = ⟪B, C⟫)
5	4	eleq1d 2419	. . 3 ⊢ (x = B → (⟪x, C⟫ ∈ A ↔ ⟪B, C⟫ ∈ A))
6	3, 5	rexsn 3768	. 2 ⊢ (∃x ∈ {B}⟪x, C⟫ ∈ A ↔ ⟪B, C⟫ ∈ A)
7	2, 6	bitri 240	1 ⊢ (C ∈ (A “k {B}) ↔ ⟪B, C⟫ ∈ A)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  {csn 3737  ⟪copk 4057   “_k cimak 4179

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 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058  df-imak 4189

This theorem is referenced by:  preaddccan2lem1  4454