Theorem iunxpf 4830

Description: Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)

Hypotheses

Ref	Expression
iunxpf.1	⊢ ℲyC
iunxpf.2	⊢ ℲzC
iunxpf.3	⊢ ℲxD
iunxpf.4	⊢ (x = ⟨y, z⟩ → C = D)

Assertion

Ref	Expression
iunxpf	⊢ ∪x ∈ (A × B)C = ∪y ∈ A ∪z ∈ B D

Distinct variable groups:   x,y,A   x,z,B,y

Allowed substitution hints:   A(z)   C(x,y,z)   D(x,y,z)

Proof of Theorem iunxpf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunxpf.1	. . . . 5 ⊢ ℲyC
2	1	nfel2 2502	. . . 4 ⊢ Ⅎy w ∈ C
3		iunxpf.2	. . . . 5 ⊢ ℲzC
4	3	nfel2 2502	. . . 4 ⊢ Ⅎz w ∈ C
5		iunxpf.3	. . . . 5 ⊢ ℲxD
6	5	nfel2 2502	. . . 4 ⊢ Ⅎx w ∈ D
7		iunxpf.4	. . . . 5 ⊢ (x = ⟨y, z⟩ → C = D)
8	7	eleq2d 2420	. . . 4 ⊢ (x = ⟨y, z⟩ → (w ∈ C ↔ w ∈ D))
9	2, 4, 6, 8	rexxpf 4829	. . 3 ⊢ (∃x ∈ (A × B)w ∈ C ↔ ∃y ∈ A ∃z ∈ B w ∈ D)
10		eliun 3974	. . 3 ⊢ (w ∈ ∪x ∈ (A × B)C ↔ ∃x ∈ (A × B)w ∈ C)
11		eliun 3974	. . . 4 ⊢ (w ∈ ∪y ∈ A ∪z ∈ B D ↔ ∃y ∈ A w ∈ ∪z ∈ B D)
12		eliun 3974	. . . . 5 ⊢ (w ∈ ∪z ∈ B D ↔ ∃z ∈ B w ∈ D)
13	12	rexbii 2640	. . . 4 ⊢ (∃y ∈ A w ∈ ∪z ∈ B D ↔ ∃y ∈ A ∃z ∈ B w ∈ D)
14	11, 13	bitri 240	. . 3 ⊢ (w ∈ ∪y ∈ A ∪z ∈ B D ↔ ∃y ∈ A ∃z ∈ B w ∈ D)
15	9, 10, 14	3bitr4i 268	. 2 ⊢ (w ∈ ∪x ∈ (A × B)C ↔ w ∈ ∪y ∈ A ∪z ∈ B D)
16	15	eqriv 2350	1 ⊢ ∪x ∈ (A × B)C = ∪y ∈ A ∪z ∈ B D

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  ∃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-13 1712  ax-14 1714  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-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-xp 4785

This theorem is referenced by: (None)