Theorem cmpcov2 22449

Description: Rewrite cmpcov 22448 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 3905	. . . . 5 ⊢ (𝑋 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑})
2		elunirab 4852	. . . . . 6 ⊢ (𝑥 ∈ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑))
3	2	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑))
4	1, 3	sylbbr 235	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑋 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑})
5		ssrab2 4009	. . . . . . 7 ⊢ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ 𝐽
6	5	unissi 4845	. . . . . 6 ⊢ ∪ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ ∪ 𝐽
7		iscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
8	6, 7	sseqtrri 3954	. . . . 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 3934	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑋 = ∪ {𝑦 ∈ 𝐽 ∣ 𝜑})
11	7	cmpcov 22448	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ {𝑦 ∈ 𝐽 ∣ 𝜑} ⊆ 𝐽 ∧ 𝑋 = ∪ {𝑦 ∈ 𝐽 ∣ 𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin)𝑋 = ∪ 𝑠)
12	5, 11	mp3an2 1447	. . 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 4002	. . . . . . . 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 642	. . . . . . 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 296	. . . . . 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 642	. . . . 5 ⊢ (((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑋 = ∪ 𝑠) ∧ 𝑠 ∈ Fin))
21		an32 642	. . . . 5 ⊢ (((𝑠 ⊆ 𝐽 ∧ 𝑠 ∈ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) ↔ ((𝑠 ⊆ 𝐽 ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
22	19, 20, 21	3bitr4i 302	. . . 4 ⊢ (((𝑠 ⊆ {𝑦 ∈ 𝐽 ∣ 𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ ((𝑠 ⊆ 𝐽 ∧ 𝑠 ∈ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
23		elfpw 9051	. . . . 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 9051	. . . . 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 302	. . 3 ⊢ ((𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin) ∧ 𝑋 = ∪ 𝑠) ↔ (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑)))
28	27	rexbii2 3175	. 2 ⊢ (∃𝑠 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ 𝜑} ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑))
29	13, 28	sylib 217	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 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  Fincfn 8691  Compccmp 22445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-cmp 22446

This theorem is referenced by:  cmpcovf  22450  bwth  22469  locfincmp  22585