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Theorem xpiundi 4817
 Description: Distributive law for cross product over indexed union. (Contributed by set.mm contributors, 26-Apr-2014.) (Revised by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundi (C × x A B) = x A (C × B)
Distinct variable group:   x,C
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem xpiundi
Dummy variables y w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom 2772 . . . 4 (w C x A y B z = w, yx A w C y B z = w, y)
2 eliun 3973 . . . . . . . 8 (y x A Bx A y B)
32anbi1i 676 . . . . . . 7 ((y x A B z = w, y) ↔ (x A y B z = w, y))
43exbii 1582 . . . . . 6 (y(y x A B z = w, y) ↔ y(x A y B z = w, y))
5 df-rex 2620 . . . . . 6 (y x A Bz = w, yy(y x A B z = w, y))
6 df-rex 2620 . . . . . . . 8 (y B z = w, yy(y B z = w, y))
76rexbii 2639 . . . . . . 7 (x A y B z = w, yx A y(y B z = w, y))
8 rexcom4 2878 . . . . . . 7 (x A y(y B z = w, y) ↔ yx A (y B z = w, y))
9 r19.41v 2764 . . . . . . . 8 (x A (y B z = w, y) ↔ (x A y B z = w, y))
109exbii 1582 . . . . . . 7 (yx A (y B z = w, y) ↔ y(x A y B z = w, y))
117, 8, 103bitri 262 . . . . . 6 (x A y B z = w, yy(x A y B z = w, y))
124, 5, 113bitr4i 268 . . . . 5 (y x A Bz = w, yx A y B z = w, y)
1312rexbii 2639 . . . 4 (w C y x A Bz = w, yw C x A y B z = w, y)
14 elxp2 4802 . . . . 5 (z (C × B) ↔ w C y B z = w, y)
1514rexbii 2639 . . . 4 (x A z (C × B) ↔ x A w C y B z = w, y)
161, 13, 153bitr4i 268 . . 3 (w C y x A Bz = w, yx A z (C × B))
17 elxp2 4802 . . 3 (z (C × x A B) ↔ w C y x A Bz = w, y)
18 eliun 3973 . . 3 (z x A (C × B) ↔ x A z (C × B))
1916, 17, 183bitr4i 268 . 2 (z (C × x A B) ↔ z x A (C × B))
2019eqriv 2350 1 (C × x A B) = x A (C × B)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∪ciun 3969  ⟨cop 4561   × cxp 4770 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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  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-sbc 3047  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-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-xp 4784 This theorem is referenced by: (None)
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