Theorem cover2 38211

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 4033	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵
3	1, 2	elpwi2 5291	. . 3 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ 𝒫 𝐵
4		nfra1 3286	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)
5	2	unissi 4874	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝐵
6	5	sseli 3932	. . . . . . 7 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 ∈ ∪ 𝐵)
7		cover2.2	. . . . . . 7 ⊢ 𝐴 = ∪ 𝐵
8	6, 7	eleqtrrdi 2873	. . . . . 6 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 ∈ 𝐴)
9		rsp 3250	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
10		elunirab 4880	. . . . . . 7 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
11	9, 10	imbitrrdi 254	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑}))
12	8, 11	impbid2 228	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → (𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴))
13	4, 12	alrimi 2248	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴))
14		dfcleq 2755	. . . 4 ⊢ (∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴))
15	13, 14	sylibr 236	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴)
16		nfrab1 3434	. . . . . . 7 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
17	16	nfeq2 2941	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}
18		eleq2 2851	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
19		rabid 3435	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ 𝜑))
20	19	simprbi 501	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝜑)
21	18, 20	biimtrdi 255	. . . . . 6 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑦 ∈ 𝑧 → 𝜑))
22	17, 21	ralrimi 3260	. . . . 5 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑦 ∈ 𝑧 𝜑)
23		unieq 4876	. . . . . . 7 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ∪ 𝑧 = ∪ {𝑦 ∈ 𝐵 ∣ 𝜑})
24	23	eqeq1d 2764	. . . . . 6 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (∪ 𝑧 = 𝐴 ↔ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴))
25	24	anbi1d 640	. . . . 5 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ (∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
26	22, 25	mpbiran2d 718	. . . 4 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴))
27	26	rspcev 3581	. . 3 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜑} ∈ 𝒫 𝐵 ∧ ∪ {𝑦 ∈ 𝐵 ∣ 𝜑} = 𝐴) → ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))
28	3, 15, 27	sylancr 596	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))
29		elpwi 4562	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝐵 → 𝑧 ⊆ 𝐵)
30		r19.29r 3126	. . . . . . . . . . 11 ⊢ ((∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦 ∈ 𝑧 𝜑) → ∃𝑦 ∈ 𝑧 (𝑥 ∈ 𝑦 ∧ 𝜑))
31	30	expcom 417	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑧 𝜑 → (∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝑧 (𝑥 ∈ 𝑦 ∧ 𝜑)))
32		ssrexv 4006	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝐵 → (∃𝑦 ∈ 𝑧 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
33	31, 32	sylan9r 516	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) → (∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
34	29, 33	sylan 589	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) → (∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
35		eleq2 2851	. . . . . . . . . 10 ⊢ (∪ 𝑧 = 𝐴 → (𝑥 ∈ ∪ 𝑧 ↔ 𝑥 ∈ 𝐴))
36	35	biimpar 481	. . . . . . . . 9 ⊢ ((∪ 𝑧 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑧)
37		eluni2 4869	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦)
38	36, 37	sylib 220	. . . . . . . 8 ⊢ ((∪ 𝑧 = 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝑧 𝑥 ∈ 𝑦)
39	34, 38	impel 513	. . . . . . 7 ⊢ (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) ∧ (∪ 𝑧 = 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
40	39	anassrs 471	. . . . . 6 ⊢ ((((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) ∧ ∪ 𝑧 = 𝐴) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
41	40	ralrimiva 3154	. . . . 5 ⊢ (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦 ∈ 𝑧 𝜑) ∧ ∪ 𝑧 = 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
42	41	anasss 470	. . . 4 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ (∀𝑦 ∈ 𝑧 𝜑 ∧ ∪ 𝑧 = 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
43	42	ancom2s 660	. . 3 ⊢ ((𝑧 ∈ 𝒫 𝐵 ∧ (∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
44	43	rexlimiva 3155	. 2 ⊢ (∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))
45	28, 44	impbii 211	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921  df-pw 4557  df-uni 4866

This theorem is referenced by:  cover2g  38212