Theorem opkelimagek 4272

Description: Membership in the Kuratowski image functor. (Contributed by SF, 20-Jan-2015.)

Hypotheses

Ref	Expression
opkelimagek.1	⊢ A ∈ V
opkelimagek.2	⊢ B ∈ V

Assertion

Ref	Expression
opkelimagek	⊢ (⟪A, B⟫ ∈ ImagekC ↔ B = (C “k A))

Proof of Theorem opkelimagek

Step	Hyp	Ref	Expression
1		opkelimagek.1	. 2 ⊢ A ∈ V
2		opkelimagek.2	. 2 ⊢ B ∈ V
3		opkelimagekg 4271	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (⟪A, B⟫ ∈ ImagekC ↔ B = (C “k A)))
4	1, 2, 3	mp2an 653	1 ⊢ (⟪A, B⟫ ∈ ImagekC ↔ B = (C “k A))

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟪copk 4057   “_k cimak 4179  Image_kcimagek 4182

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-nin 4078  ax-sn 4087

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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-sik 4192  df-ssetk 4193  df-imagek 4194

This theorem is referenced by:  preaddccan2lem1  4454  ltfinex  4464  evenodddisjlem1  4515  dfphi2  4569  dfop2lem1  4573  dfop2  4575  dfproj12  4576  phialllem1  4616  setconslem1  4731  setconslem2  4732  dfswap2  4741