Theorem rabasiun 4459

Description: A class abstraction with a restricted existential quantification expressed as indexed union. (Contributed by Alexander van der Vekens, 29-Jul-2018.)

Assertion

Ref	Expression
rabasiun	⊢ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜑} = ∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜑}

Distinct variable groups:   𝑥,𝑋,𝑦   𝑥,𝑌

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑌(𝑦)

Proof of Theorem rabasiun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑧𝑋
2	1	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑋
3		nfv 1830	. . . . 5 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝑌 𝜑
4	2, 3	nfan 1816	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 𝜑)
5		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝑋
6	5	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝑋
7		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝑌
8		nfs1v 2425	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
9	7, 8	nfrex 2990	. . . . 5 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑
10	6, 9	nfan 1816	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑)
11		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑋 ↔ 𝑧 ∈ 𝑋))
12		sbequ12 2097	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13	12	rexbidv 3034	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝑌 𝜑 ↔ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑))
14	11, 13	anbi12d 743	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 𝜑) ↔ (𝑧 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑)))
15	4, 10, 14	cbvab 2733	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 𝜑)} = {𝑧 ∣ (𝑧 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑)}
16		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
17	16, 5, 8, 12	elrabf 3329	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝑋 ∣ 𝜑} ↔ (𝑧 ∈ 𝑋 ∧ [𝑧 / 𝑥]𝜑))
18	17	rexbii 3023	. . . . 5 ⊢ (∃𝑦 ∈ 𝑌 𝑧 ∈ {𝑥 ∈ 𝑋 ∣ 𝜑} ↔ ∃𝑦 ∈ 𝑌 (𝑧 ∈ 𝑋 ∧ [𝑧 / 𝑥]𝜑))
19		r19.42v 3073	. . . . 5 ⊢ (∃𝑦 ∈ 𝑌 (𝑧 ∈ 𝑋 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑))
20	18, 19	bitr2i 264	. . . 4 ⊢ ((𝑧 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑) ↔ ∃𝑦 ∈ 𝑌 𝑧 ∈ {𝑥 ∈ 𝑋 ∣ 𝜑})
21	20	abbii 2726	. . 3 ⊢ {𝑧 ∣ (𝑧 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 [𝑧 / 𝑥]𝜑)} = {𝑧 ∣ ∃𝑦 ∈ 𝑌 𝑧 ∈ {𝑥 ∈ 𝑋 ∣ 𝜑}}
22	15, 21	eqtri 2632	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 𝜑)} = {𝑧 ∣ ∃𝑦 ∈ 𝑌 𝑧 ∈ {𝑥 ∈ 𝑋 ∣ 𝜑}}
23		df-rab 2905	. 2 ⊢ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑌 𝜑)}
24		df-iun 4457	. 2 ⊢ ∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜑} = {𝑧 ∣ ∃𝑦 ∈ 𝑌 𝑧 ∈ {𝑥 ∈ 𝑋 ∣ 𝜑}}
25	22, 23, 24	3eqtr4i 2642	1 ⊢ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜑} = ∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜑}

Syntax hints:   ∧ wa 383   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {crab 2900  ∪ ciun 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-iun 4457

This theorem is referenced by:  hashrabrex  14396