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Theorem dfpw12 4301
Description: Alternate expression for unit power classes. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
dfpw12 1A = ( SIk (A ×k A) “k V)

Proof of Theorem dfpw12
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpw1 4144 . . 3 (x 1Ay A x = {y})
2 vex 2862 . . . . 5 x V
32elimakv 4260 . . . 4 (x ( SIk (A ×k A) “k V) ↔ zz, x SIk (A ×k A))
4 vex 2862 . . . . . . 7 z V
5 opkelsikg 4264 . . . . . . 7 ((z V x V) → (⟪z, x SIk (A ×k A) ↔ wy(z = {w} x = {y} w, y (A ×k A))))
64, 2, 5mp2an 653 . . . . . 6 (⟪z, x SIk (A ×k A) ↔ wy(z = {w} x = {y} w, y (A ×k A)))
76exbii 1582 . . . . 5 (zz, x SIk (A ×k A) ↔ zwy(z = {w} x = {y} w, y (A ×k A)))
8 exrot3 1744 . . . . 5 (yzw(z = {w} x = {y} w, y (A ×k A)) ↔ zwy(z = {w} x = {y} w, y (A ×k A)))
97, 8bitr4i 243 . . . 4 (zz, x SIk (A ×k A) ↔ yzw(z = {w} x = {y} w, y (A ×k A)))
10 df-3an 936 . . . . . . . . 9 ((z = {w} x = {y} w, y (A ×k A)) ↔ ((z = {w} x = {y}) w, y (A ×k A)))
11 vex 2862 . . . . . . . . . . 11 w V
12 vex 2862 . . . . . . . . . . 11 y V
1311, 12opkelxpk 4248 . . . . . . . . . 10 (⟪w, y (A ×k A) ↔ (w A y A))
1413anbi2i 675 . . . . . . . . 9 (((z = {w} x = {y}) w, y (A ×k A)) ↔ ((z = {w} x = {y}) (w A y A)))
15 an4 797 . . . . . . . . 9 (((z = {w} x = {y}) (w A y A)) ↔ ((z = {w} w A) (x = {y} y A)))
1610, 14, 153bitri 262 . . . . . . . 8 ((z = {w} x = {y} w, y (A ×k A)) ↔ ((z = {w} w A) (x = {y} y A)))
17162exbii 1583 . . . . . . 7 (zw(z = {w} x = {y} w, y (A ×k A)) ↔ zw((z = {w} w A) (x = {y} y A)))
18 19.41vv 1902 . . . . . . 7 (zw((z = {w} w A) (x = {y} y A)) ↔ (zw(z = {w} w A) (x = {y} y A)))
19 sneq 3744 . . . . . . . . . . . 12 (w = y → {w} = {y})
20 eqeq12 2365 . . . . . . . . . . . 12 ((z = x {w} = {y}) → (z = {w} ↔ x = {y}))
2119, 20sylan2 460 . . . . . . . . . . 11 ((z = x w = y) → (z = {w} ↔ x = {y}))
22 eleq1 2413 . . . . . . . . . . . 12 (w = y → (w Ay A))
2322adantl 452 . . . . . . . . . . 11 ((z = x w = y) → (w Ay A))
2421, 23anbi12d 691 . . . . . . . . . 10 ((z = x w = y) → ((z = {w} w A) ↔ (x = {y} y A)))
252, 12, 24spc2ev 2947 . . . . . . . . 9 ((x = {y} y A) → zw(z = {w} w A))
2625pm4.71ri 614 . . . . . . . 8 ((x = {y} y A) ↔ (zw(z = {w} w A) (x = {y} y A)))
27 ancom 437 . . . . . . . 8 ((x = {y} y A) ↔ (y A x = {y}))
2826, 27bitr3i 242 . . . . . . 7 ((zw(z = {w} w A) (x = {y} y A)) ↔ (y A x = {y}))
2917, 18, 283bitri 262 . . . . . 6 (zw(z = {w} x = {y} w, y (A ×k A)) ↔ (y A x = {y}))
3029exbii 1582 . . . . 5 (yzw(z = {w} x = {y} w, y (A ×k A)) ↔ y(y A x = {y}))
31 df-rex 2620 . . . . 5 (y A x = {y} ↔ y(y A x = {y}))
3230, 31bitr4i 243 . . . 4 (yzw(z = {w} x = {y} w, y (A ×k A)) ↔ y A x = {y})
333, 9, 323bitri 262 . . 3 (x ( SIk (A ×k A) “k V) ↔ y A x = {y})
341, 33bitr4i 243 . 2 (x 1Ax ( SIk (A ×k A) “k V))
3534eqriv 2350 1 1A = ( SIk (A ×k A) “k V)
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358   w3a 934  wex 1541   = wceq 1642   wcel 1710  wrex 2615  Vcvv 2859  {csn 3737  copk 4057  1cpw1 4135   ×k cxpk 4174  k cimak 4179   SIk csik 4181
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-rex 2620  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-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-imak 4189  df-sik 4192
This theorem is referenced by:  pw1exg  4302
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