Theorem iuniin 4893

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin	⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuniin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 3283	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		eliin 4886	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
3	2	elv 3446	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
4	3	rexbii 3210	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
5		eliun 4885	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
6	5	ralbii 3133	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
7	1, 4, 6	3imtr4i 295	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 → ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
8		eliun 4885	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶)
9		eliin 4886	. . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
10	9	elv 3446	. . 3 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
11	7, 8, 10	3imtr4i 295	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 → 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
12	11	ssriv 3919	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   ↔ wb 209   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∪ ciun 4881  ∩ ciin 4882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-iun 4883  df-iin 4884

This theorem is referenced by: (None)