Theorem iuniin 3974

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 2637	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		vex 2802	. . . . . 6 ⊢ 𝑧 ∈ V
3		eliin 3969	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
5	4	rexbii 2537	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
6		eliun 3968	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
7	6	ralbii 2536	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
8	1, 5, 7	3imtr4i 201	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 → ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
9		eliun 3968	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶)
10		eliin 3969	. . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
11	2, 10	ax-mp 5	. . 3 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
12	8, 9, 11	3imtr4i 201	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 → 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
13	12	ssriv 3228	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   ↔ wb 105   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ⊆ wss 3197  ∪ ciun 3964  ∩ ciin 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-iun 3966  df-iin 3967

This theorem is referenced by: (None)