Theorem dfiunv2 4032

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 3998	. . . 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 4014	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}
4		df-iun 3998	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}}
5		vex 2818	. . . . 5 ⊢ 𝑧 ∈ V
6		eleq1 2297	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
7	6	rexbidv 2545	. . . . 5 ⊢ (𝑤 = 𝑧 → (∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
8	5, 7	elab 2964	. . . 4 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
9	8	rexbii 2551	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
10	9	abbii 2350	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}
11	3, 4, 10	3eqtri 2259	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶}

Syntax hints:   = wceq 1398   ∈ wcel 2205  {cab 2220  ∃wrex 2523  ∪ ciun 3996

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227  df-iun 3998

This theorem is referenced by: (None)