Theorem dfiunv2 5031

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 4992	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}
2	1	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶})
3	2	iuneq2i 5011	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}
4		df-iun 4992	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}}
5		vex 3472	. . . . 5 ⊢ 𝑧 ∈ V
6		eleq1w 2810	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
7	6	rexbidv 3172	. . . . 5 ⊢ (𝑤 = 𝑧 → (∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
8	5, 7	elab 3663	. . . 4 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
9	8	rexbii 3088	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
10	9	abbii 2796	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}
11	3, 4, 10	3eqtri 2758	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}

Syntax hints:   = wceq 1533   ∈ wcel 2098  {cab 2703  ∃wrex 3064  ∪ ciun 4990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-iun 4992

This theorem is referenced by:  wspniunwspnon  29686  fusgr2wsp2nb  30096