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Theorem dfidk2 4313
 Description: Definition of Ik in terms of Sk. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
dfidk2 Ik = ( Skk Sk )

Proof of Theorem dfidk2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idkssvvk 4281 . 2 Ik (V ×k V)
2 inss1 3475 . . 3 ( Skk Sk ) Sk
3 ssetkssvvk 4278 . . 3 Sk (V ×k V)
42, 3sstri 3281 . 2 ( Skk Sk ) (V ×k V)
5 eqss 3287 . . 3 (x = y ↔ (x y y x))
6 vex 2862 . . . 4 x V
7 vex 2862 . . . 4 y V
8 opkelidkg 4274 . . . 4 ((x V y V) → (⟪x, y Ikx = y))
96, 7, 8mp2an 653 . . 3 (⟪x, y Ikx = y)
10 elin 3219 . . . 4 (⟪x, y ( Skk Sk ) ↔ (⟪x, y Sk x, y k Sk ))
11 opkelssetkg 4268 . . . . . 6 ((x V y V) → (⟪x, y Skx y))
126, 7, 11mp2an 653 . . . . 5 (⟪x, y Skx y)
136, 7opkelcnvk 4250 . . . . . 6 (⟪x, y k Sk ↔ ⟪y, x Sk )
14 opkelssetkg 4268 . . . . . . 7 ((y V x V) → (⟪y, x Sky x))
157, 6, 14mp2an 653 . . . . . 6 (⟪y, x Sky x)
1613, 15bitri 240 . . . . 5 (⟪x, y k Sky x)
1712, 16anbi12i 678 . . . 4 ((⟪x, y Sk x, y k Sk ) ↔ (x y y x))
1810, 17bitri 240 . . 3 (⟪x, y ( Skk Sk ) ↔ (x y y x))
195, 9, 183bitr4i 268 . 2 (⟪x, y Ik ↔ ⟪x, y ( Skk Sk ))
201, 4, 19eqrelkriiv 4213 1 Ik = ( Skk Sk )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ⟪copk 4057   ×k cxpk 4174  ◡kccnvk 4175   Sk cssetk 4183   Ik cidk 4184 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  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-xpk 4185  df-cnvk 4186  df-ssetk 4193  df-idk 4195 This theorem is referenced by:  idkex  4314
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