Theorem xpiundi 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
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.

Step	Hyp	Ref	Expression
1		rexcom 2773	. . . 4 ⊢ (∃w ∈ C ∃x ∈ A ∃y ∈ B z = ⟨w, y⟩ ↔ ∃x ∈ A ∃w ∈ C ∃y ∈ B z = ⟨w, y⟩)
2		eliun 3974	. . . . . . . 8 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
3	2	anbi1i 676	. . . . . . 7 ⊢ ((y ∈ ∪x ∈ A B ∧ z = ⟨w, y⟩) ↔ (∃x ∈ A y ∈ B ∧ z = ⟨w, y⟩))
4	3	exbii 1582	. . . . . 6 ⊢ (∃y(y ∈ ∪x ∈ A B ∧ z = ⟨w, y⟩) ↔ ∃y(∃x ∈ A y ∈ B ∧ z = ⟨w, y⟩))
5		df-rex 2621	. . . . . 6 ⊢ (∃y ∈ ∪ x ∈ A Bz = ⟨w, y⟩ ↔ ∃y(y ∈ ∪x ∈ A B ∧ z = ⟨w, y⟩))
6		df-rex 2621	. . . . . . . 8 ⊢ (∃y ∈ B z = ⟨w, y⟩ ↔ ∃y(y ∈ B ∧ z = ⟨w, y⟩))
7	6	rexbii 2640	. . . . . . 7 ⊢ (∃x ∈ A ∃y ∈ B z = ⟨w, y⟩ ↔ ∃x ∈ A ∃y(y ∈ B ∧ z = ⟨w, y⟩))
8		rexcom4 2879	. . . . . . 7 ⊢ (∃x ∈ A ∃y(y ∈ B ∧ z = ⟨w, y⟩) ↔ ∃y∃x ∈ A (y ∈ B ∧ z = ⟨w, y⟩))
9		r19.41v 2765	. . . . . . . 8 ⊢ (∃x ∈ A (y ∈ B ∧ z = ⟨w, y⟩) ↔ (∃x ∈ A y ∈ B ∧ z = ⟨w, y⟩))
10	9	exbii 1582	. . . . . . 7 ⊢ (∃y∃x ∈ A (y ∈ B ∧ z = ⟨w, y⟩) ↔ ∃y(∃x ∈ A y ∈ B ∧ z = ⟨w, y⟩))
11	7, 8, 10	3bitri 262	. . . . . 6 ⊢ (∃x ∈ A ∃y ∈ B z = ⟨w, y⟩ ↔ ∃y(∃x ∈ A y ∈ B ∧ z = ⟨w, y⟩))
12	4, 5, 11	3bitr4i 268	. . . . 5 ⊢ (∃y ∈ ∪ x ∈ A Bz = ⟨w, y⟩ ↔ ∃x ∈ A ∃y ∈ B z = ⟨w, y⟩)
13	12	rexbii 2640	. . . 4 ⊢ (∃w ∈ C ∃y ∈ ∪ x ∈ A Bz = ⟨w, y⟩ ↔ ∃w ∈ C ∃x ∈ A ∃y ∈ B z = ⟨w, y⟩)
14		elxp2 4803	. . . . 5 ⊢ (z ∈ (C × B) ↔ ∃w ∈ C ∃y ∈ B z = ⟨w, y⟩)
15	14	rexbii 2640	. . . 4 ⊢ (∃x ∈ A z ∈ (C × B) ↔ ∃x ∈ A ∃w ∈ C ∃y ∈ B z = ⟨w, y⟩)
16	1, 13, 15	3bitr4i 268	. . 3 ⊢ (∃w ∈ C ∃y ∈ ∪ x ∈ A Bz = ⟨w, y⟩ ↔ ∃x ∈ A z ∈ (C × B))
17		elxp2 4803	. . 3 ⊢ (z ∈ (C × ∪x ∈ A B) ↔ ∃w ∈ C ∃y ∈ ∪ x ∈ A Bz = ⟨w, y⟩)
18		eliun 3974	. . 3 ⊢ (z ∈ ∪x ∈ A (C × B) ↔ ∃x ∈ A z ∈ (C × B))
19	16, 17, 18	3bitr4i 268	. 2 ⊢ (z ∈ (C × ∪x ∈ A B) ↔ z ∈ ∪x ∈ A (C × B))
20	19	eqriv 2350	1 ⊢ (C × ∪x ∈ A B) = ∪x ∈ A (C × B)

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∪ciun 3970  ⟨cop 4562   × cxp 4771

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-xp 4785

This theorem is referenced by: (None)