Theorem elimak 4259

Description: Membership in a Kuratowski image. (Contributed by SF, 13-Jan-2015.)

Hypothesis

Ref	Expression
elimak.1	⊢ C ∈ V

Assertion

Ref	Expression
elimak	⊢ (C ∈ (A “k B) ↔ ∃y ∈ B ⟪y, C⟫ ∈ A)

Distinct variable groups:   y,A   y,B   y,C

Proof of Theorem elimak

Step	Hyp	Ref	Expression
1		elimak.1	. 2 ⊢ C ∈ V
2		elimakg 4257	. 2 ⊢ (C ∈ V → (C ∈ (A “k B) ↔ ∃y ∈ B ⟪y, C⟫ ∈ A))
3	1, 2	ax-mp 5	1 ⊢ (C ∈ (A “k B) ↔ ∃y ∈ B ⟪y, C⟫ ∈ A)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  ⟪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-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-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:  opkelimagekg  4271  imacok  4282  elimaksn  4283  dfimak2  4298  dfuni3  4315  dfint3  4318  ndisjrelk  4323  dfpw2  4327  dfaddc2  4381  dfnnc2  4395  nnc0suc  4412  nncaddccl  4419  nnsucelrlem1  4424  nndisjeq  4429  preaddccan2lem1  4454  ltfinex  4464  ltfintrilem1  4465  ssfin  4470  eqpwrelk  4478  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelk  4524  sfintfinlem1  4531  tfinnnlem1  4533  spfinex  4537  vfinspss  4551  vfinspclt  4552  vfinncsp  4554  dfphi2  4569  dfop2lem1  4573  dfop2lem2  4574  dfop2  4575  dfproj12  4576  dfproj22  4577  phialllem1  4616  setconslem2  4732  setconslem3  4733  setconslem4  4734  setconslem6  4736  setconslem7  4737  df1st2  4738  dfswap2  4741  dfima2  4745