Theorem elimak 4260

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 4258	. 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 2616  Vcvv 2860  ⟪copk 4058   “_k cimak 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-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-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:  opkelimagekg  4272  imacok  4283  elimaksn  4284  dfimak2  4299  dfuni3  4316  dfint3  4319  ndisjrelk  4324  dfpw2  4328  dfaddc2  4382  dfnnc2  4396  nnc0suc  4413  nncaddccl  4420  nnsucelrlem1  4425  nndisjeq  4430  preaddccan2lem1  4455  ltfinex  4465  ltfintrilem1  4466  ssfin  4471  eqpwrelk  4479  eqpw1relk  4480  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelk  4525  sfintfinlem1  4532  tfinnnlem1  4534  spfinex  4538  vfinspss  4552  vfinspclt  4553  vfinncsp  4555  dfphi2  4570  dfop2lem1  4574  dfop2lem2  4575  dfop2  4576  dfproj12  4577  dfproj22  4578  phialllem1  4617  setconslem2  4733  setconslem3  4734  setconslem4  4735  setconslem6  4737  setconslem7  4738  df1st2  4739  dfswap2  4742  dfima2  4746