Theorem dfiunv2 3714

Description: Define double indexed union. (Contributed by FL, 6-Nov-2013.)

Assertion

Ref	Expression
dfiunv2	⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}

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

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

Proof of Theorem dfiunv2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3680	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}
2	1	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶})
3	2	iuneq2i 3696	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}
4		df-iun 3680	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}}
5		vex 2604	. . . . 5 ⊢ 𝑧 ∈ V
6		eleq1 2141	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
7	6	rexbidv 2369	. . . . 5 ⊢ (𝑤 = 𝑧 → (∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
8	5, 7	elab 2738	. . . 4 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
9	8	rexbii 2373	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
10	9	abbii 2194	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}
11	3, 4, 10	3eqtri 2105	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}

Syntax hints:   = wceq 1284   ∈ wcel 1433  {cab 2067  ∃wrex 2349  ∪ ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-iun 3680

This theorem is referenced by: (None)