Theorem cover2 38253

Description: Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)

Hypotheses

Ref	Expression
cover2.1	⊢ 𝐵 ∈ V
cover2.2	⊢ 𝐴 = ∪ 𝐵

Assertion

Ref	Expression
cover2	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))

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

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem cover2

Step	Hyp	Ref	Expression
1		cover2.1	. . . 4 ⊢ 𝐵 ∈ V
2		ssrab2 4042	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵
3	1, 2	elpwi2 5306	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ 𝒫 𝐵
4		nfra1 3295	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)
5	2	unissi 4885	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝐵
6	5	sseli 3941	. . . . . . 7 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 ∈ ∪ 𝐵)
7		cover2.2	. . . . . . 7 ⊢ 𝐴 = ∪ 𝐵
8	6, 7	eleqtrrdi 2880	. . . . . 6 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 ∈ 𝐴)
9		rsp 3259	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
10		elunirab 4891	. . . . . . 7 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
11	9, 10	imbitrrdi 255	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑}))
12	8, 11	impbid2 229	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴))
13	4, 12	alrimi 2255	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴))
14		dfcleq 2762	. . . 4 ⊢ (∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴))
15	13, 14	sylibr 237	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴)
16		nfrab1 3443	. . . . . . 7 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
17	16	nfeq2 2948	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}
18		eleq2 2858	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
19		rabid 3444	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ 𝜑))
20	19	simprbi 502	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝜑)
21	18, 20	biimtrdi 256	. . . . . 6 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑦 ∈ 𝑧 → 𝜑))
22	17, 21	ralrimi 3269	. . . . 5 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑦 ∈ 𝑧 𝜑)
23		unieq 4887	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ∪ 𝑧 = ∪ {𝑦 ∈ 𝐵 ∣ 𝜑})
24	23	eqeq1d 2771	. . . . . 6 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (∪ 𝑧 = 𝐴 ↔ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴))
25	24	anbi1d 642	. . . . 5 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ (∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
26	22, 25	mpbiran2d 720	. . . 4 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴))
27	26	rspcev 3590	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜑} ∈ 𝒫 𝐵 ∧ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴) → ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))
28	3, 15, 27	sylancr 598	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))
29		elpwi 4574	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝐵 → 𝑧 ⊆ 𝐵)
30		r19.29r 3135	. . . . . . . . . . 11 ⊢ ((∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦 ∈ 𝑧 𝜑) → ∃𝑦 ∈ 𝑧 (𝑥 ∈ 𝑦 ∧ 𝜑))
31	30	expcom 418	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑧 𝜑 → (∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝑧 (𝑥 ∈ 𝑦 ∧ 𝜑)))
32		ssrexv 4015	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝐵 → (∃𝑦 ∈ 𝑧 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
33	31, 32	sylan9r 517	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) → (∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
34	29, 33	sylan 591	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) → (∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
35		eleq2 2858	. . . . . . . . . 10 ⊢ (∪ 𝑧 = 𝐴 → (𝑥 ∈ ∪ 𝑧 ↔ 𝑥 ∈ 𝐴))
36	35	biimpar 482	. . . . . . . . 9 ⊢ ((∪ 𝑧 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑧)
37		eluni2 4880	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦)
38	36, 37	sylib 221	. . . . . . . 8 ⊢ ((∪ 𝑧 = 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦)
39	34, 38	impel 514	. . . . . . 7 ⊢ (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) ∧ (∪ 𝑧 = 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
40	39	anassrs 472	. . . . . 6 ⊢ ((((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) ∧ ∪ 𝑧 = 𝐴) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
41	40	ralrimiva 3163	. . . . 5 ⊢ (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) ∧ ∪ 𝑧 = 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
42	41	anasss 471	. . . 4 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ (∀𝑦 ∈ 𝑧 𝜑 ∧ ∪ 𝑧 = 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
43	42	ancom2s 662	. . 3 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ (∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
44	43	rexlimiva 3164	. 2 ⊢ (∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
45	28, 44	impbii 212	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4569  df-uni 4877

This theorem is referenced by:  cover2g  38254