Theorem rabiun 33382

Description: Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)

Assertion

Ref	Expression
rabiun	⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝜑}

Distinct variable groups:   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦

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

Proof of Theorem rabiun

Step	Hyp	Ref	Expression
1		eliun 4524	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵)
2	1	anbi1i 731	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝜑))
3		r19.41v 3089	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝜑))
4	2, 3	bitr4i 267	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑))
5	4	abbii 2739	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑)}
6		df-rab 2921	. . 3 ⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑)}
7		iunab 4566	. . 3 ⊢ ∪ 𝑦 ∈ 𝐴 {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑)}
8	5, 6, 7	3eqtr4i 2654	. 2 ⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
9		df-rab 2921	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
10	9	a1i 11	. . 3 ⊢ (𝑦 ∈ 𝐴 → {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
11	10	iuneq2i 4539	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
12	8, 11	eqtr4i 2647	1 ⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝜑}

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  {crab 2916  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  itg2addnclem2  33462