Theorem cmpcov2 23419

Description: Rewrite cmpcov 23418 for the cover {𝑦 ∈ 𝐽 ∣ 𝜑}. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
iscmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cmpcov2	⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑))

Distinct variable groups:   𝑥,𝑠,𝑦,𝐽   𝜑,𝑠,𝑥   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝑋(𝑦,𝑠)

Proof of Theorem cmpcov2

Step	Hyp	Ref	Expression
1		dfss3 3997	. . . . 5 ⊢ (𝑋 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑})
2		elunirab 4946	. . . . . 6 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑))
3	2	ralbii 3099	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑))
4	1, 3	sylbbr 236	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑋 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑})
5		ssrab2 4103	. . . . . . 7 ⊢ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ 𝐽
6	5	unissi 4940	. . . . . 6 ⊢ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ ∪ 𝐽
7		iscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
8	6, 7	sseqtrri 4046	. . . . 5 ⊢ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ 𝑋
9	8	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑) → ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ 𝑋)
10	4, 9	eqssd 4026	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑋 = ∪ {𝑦 ∈ 𝐽 ∣ 𝜑})
11	7	cmpcov 23418	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ 𝐽 ∧ 𝑋 = ∪ {𝑦 ∈ 𝐽 ∣ 𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin)𝑋 = ∪ 𝑠)
12	5, 11	mp3an2 1449	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑋 = ∪ {𝑦 ∈ 𝐽 ∣ 𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin)𝑋 = ∪ 𝑠)
13	10, 12	sylan2 592	. 2 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin)𝑋 = ∪ 𝑠)
14		ssrab 4096	. . . . . . . 8 ⊢ (𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ (𝑠 ⊆ 𝐽 ∧ ∀𝑦 ∈ 𝑠 𝜑))
15	14	anbi1i 623	. . . . . . 7 ⊢ ((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ 𝐽 ∧ ∀𝑦 ∈ 𝑠 𝜑) ∧ 𝑋 = ∪ 𝑠))
16		an32 645	. . . . . . 7 ⊢ (((𝑠 ⊆ 𝐽 ∧ ∀𝑦 ∈ 𝑠 𝜑) ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑠) ∧ ∀𝑦 ∈ 𝑠 𝜑))
17		anass 468	. . . . . . 7 ⊢ (((𝑠 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑠) ∧ ∀𝑦 ∈ 𝑠 𝜑) ↔ (𝑠 ⊆ 𝐽 ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
18	15, 16, 17	3bitri 297	. . . . . 6 ⊢ ((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑋 = ∪ 𝑠) ↔ (𝑠 ⊆ 𝐽 ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
19	18	anbi1i 623	. . . . 5 ⊢ (((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑋 = ∪ 𝑠) ∧ 𝑠 ∈ Fin) ↔ ((𝑠 ⊆ 𝐽 ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
20		an32 645	. . . . 5 ⊢ (((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑋 = ∪ 𝑠) ∧ 𝑠 ∈ Fin))
21		an32 645	. . . . 5 ⊢ (((𝑠 ⊆ 𝐽 ∧ 𝑠 ∈ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) ↔ ((𝑠 ⊆ 𝐽 ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
22	19, 20, 21	3bitr4i 303	. . . 4 ⊢ (((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ 𝐽 ∧ 𝑠 ∈ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
23		elfpw 9424	. . . . 5 ⊢ (𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin) ↔ (𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑠 ∈ Fin))
24	23	anbi1i 623	. . . 4 ⊢ ((𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = ∪ 𝑠))
25		elfpw 9424	. . . . 5 ⊢ (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑠 ⊆ 𝐽 ∧ 𝑠 ∈ Fin))
26	25	anbi1i 623	. . . 4 ⊢ ((𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) ↔ ((𝑠 ⊆ 𝐽 ∧ 𝑠 ∈ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
27	22, 24, 26	3bitr4i 303	. . 3 ⊢ ((𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
28	27	rexbii2 3096	. 2 ⊢ (∃𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑))
29	13, 28	sylib 218	1 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  Fincfn 9003  Compccmp 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624  df-uni 4932  df-cmp 23416

This theorem is referenced by:  cmpcovf  23420  bwth  23439  locfincmp  23555