Theorem cflim3 10259

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 6423	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
2		cflim3.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	elon 6372	. . . 4 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
4	1, 3	sylibr 233	. . 3 ⊢ (Lim 𝐴 → 𝐴 ∈ On)
5		cfval 10244	. . 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 6903	. . . 4 ⊢ (card‘𝑥) ∈ V
8	7	dfiin2 5036	. . 3 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9		df-rex 3069	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)))
10		ancom 459	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}))
11		rabid 3450	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
12		velpw 4606	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	12	anbi1i 622	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴))
14		coflim 10258	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝑥 = 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))
15	14	pm5.32da 577	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
16	13, 15	bitrid 282	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
17	11, 16	bitrid 282	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
18	17	anbi2d 627	. . . . . . . 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 282	. . . . . . 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 1922	. . . . . 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 282	. . . . 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 2799	. . . 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 4954	. . 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 2783	. 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 2770	1 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2707  ∀wral 3059  ∃wrex 3068  {crab 3430  Vcvv 3472   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  ∩ cint 4949  ∩ ciin 4997  Ord word 6362  Oncon0 6363  Lim wlim 6364  ‘cfv 6542  cardccrd 9932  cfccf 9934

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-ord 6366  df-on 6367  df-lim 6368  df-iota 6494  df-fun 6544  df-fv 6550  df-cf 9938

This theorem is referenced by:  cflim2  10260  cfss  10262  cfslb  10263