Theorem iuniin 3748

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 2480	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		vex 2625	. . . . . 6 ⊢ 𝑧 ∈ V
3		eliin 3743	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
4	2, 3	ax-mp 7	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
5	4	rexbii 2386	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
6		eliun 3742	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
7	6	ralbii 2385	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
8	1, 5, 7	3imtr4i 200	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 → ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
9		eliun 3742	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶)
10		eliin 3743	. . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
11	2, 10	ax-mp 7	. . 3 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
12	8, 9, 11	3imtr4i 200	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 → 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
13	12	ssriv 3032	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   ↔ wb 104   ∈ wcel 1439  ∀wral 2360  ∃wrex 2361  Vcvv 2622   ⊆ wss 3002  ∪ ciun 3738  ∩ ciin 3739

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-in 3008  df-ss 3015  df-iun 3740  df-iin 3741

This theorem is referenced by: (None)