Theorem iunxpf 5787

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 2886	. . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶
3		iunxpf.2	. . . . 5 ⊢ Ⅎ𝑧𝐶
4	3	nfcri 2886	. . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶
5		iunxpf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
6	5	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
7		iunxpf.4	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
8	7	eleq2d 2817	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷))
9	2, 4, 6, 8	rexxpf 5786	. . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
10		eliun 4943	. . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶)
11		eliun 4943	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷)
12		eliun 4943	. . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
13	12	rexbii 3079	. . . 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 2728	1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  ∃wrex 3056  ⟨cop 4579  ∪ ciun 4939   × cxp 5612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-iun 4941  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by:  dfmpo  8032  pzriprnglem11  21428