Theorem iunxpf 5828

Description: Indexed union on a Cartesian product 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	nfcri 2890	. . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶
3		iunxpf.2	. . . . 5 ⊢ Ⅎ𝑧𝐶
4	3	nfcri 2890	. . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶
5		iunxpf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
6	5	nfcri 2890	. . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
7		iunxpf.4	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
8	7	eleq2d 2820	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷))
9	2, 4, 6, 8	rexxpf 5827	. . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
10		eliun 4971	. . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶)
11		eliun 4971	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷)
12		eliun 4971	. . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
13	12	rexbii 3083	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
14	11, 13	bitri 275	. . 3 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
15	9, 10, 14	3bitr4i 303	. 2 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷)
16	15	eqriv 2732	1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2883  ∃wrex 3060  ⟨cop 4607  ∪ ciun 4967   × cxp 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-iun 4969  df-opab 5182  df-xp 5660  df-rel 5661

This theorem is referenced by:  dfmpo  8099  pzriprnglem11  21450