Theorem cflim3 10302

Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Hypothesis

Ref	Expression
cflim3.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cflim3	⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem cflim3

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limord 6444	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
2		cflim3.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	elon 6393	. . . 4 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
4	1, 3	sylibr 234	. . 3 ⊢ (Lim 𝐴 → 𝐴 ∈ On)
5		cfval 10287	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
6	4, 5	syl 17	. 2 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
7		fvex 6919	. . . 4 ⊢ (card‘𝑥) ∈ V
8	7	dfiin2 5034	. . 3 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9		df-rex 3071	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)))
10		ancom 460	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}))
11		rabid 3458	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
12		velpw 4605	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	12	anbi1i 624	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴))
14		coflim 10301	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝑥 = 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))
15	14	pm5.32da 579	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
16	13, 15	bitrid 283	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
17	11, 16	bitrid 283	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
18	17	anbi2d 630	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
19	10, 18	bitrid 283	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
20	19	exbidv 1921	. . . . . 6 ⊢ (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
21	9, 20	bitrid 283	. . . . 5 ⊢ (Lim 𝐴 → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
22	21	abbidv 2808	. . . 4 ⊢ (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
23	22	inteqd 4951	. . 3 ⊢ (Lim 𝐴 → ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
24	8, 23	eqtr2id 2790	. 2 ⊢ (Lim 𝐴 → ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))} = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
25	6, 24	eqtrd 2777	1 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ∩ cint 4946  ∩ ciin 4992  Ord word 6383  Oncon0 6384  Lim wlim 6385  ‘cfv 6561  cardccrd 9975  cfccf 9977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-iota 6514  df-fun 6563  df-fv 6569  df-cf 9981

This theorem is referenced by:  cflim2  10303  cfss  10305  cfslb  10306