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Theorem xpiundir 4818
 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
xpiundir (x A B × C) = x A (B × C)
Distinct variable group:   x,C
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem xpiundir
Dummy variables y w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2878 . . . . 5 (x A y(y B w C z = y, w) ↔ yx A (y B w C z = y, w))
2 df-rex 2620 . . . . . 6 (y B w C z = y, wy(y B w C z = y, w))
32rexbii 2639 . . . . 5 (x A y B w C z = y, wx A y(y B w C z = y, w))
4 eliun 3973 . . . . . . . 8 (y x A Bx A y B)
54anbi1i 676 . . . . . . 7 ((y x A B w C z = y, w) ↔ (x A y B w C z = y, w))
6 r19.41v 2764 . . . . . . 7 (x A (y B w C z = y, w) ↔ (x A y B w C z = y, w))
75, 6bitr4i 243 . . . . . 6 ((y x A B w C z = y, w) ↔ x A (y B w C z = y, w))
87exbii 1582 . . . . 5 (y(y x A B w C z = y, w) ↔ yx A (y B w C z = y, w))
91, 3, 83bitr4ri 269 . . . 4 (y(y x A B w C z = y, w) ↔ x A y B w C z = y, w)
10 df-rex 2620 . . . 4 (y x A Bw C z = y, wy(y x A B w C z = y, w))
11 elxp2 4802 . . . . 5 (z (B × C) ↔ y B w C z = y, w)
1211rexbii 2639 . . . 4 (x A z (B × C) ↔ x A y B w C z = y, w)
139, 10, 123bitr4i 268 . . 3 (y x A Bw C z = y, wx A z (B × C))
14 elxp2 4802 . . 3 (z (x A B × C) ↔ y x A Bw C z = y, w)
15 eliun 3973 . . 3 (z x A (B × C) ↔ x A z (B × C))
1613, 14, 153bitr4i 268 . 2 (z (x A B × C) ↔ z x A (B × C))
1716eqriv 2350 1 (x A B × C) = x A (B × C)
 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:  iunxpconst  4819
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