Theorem cflim3 10331

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 6455	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
2		cflim3.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	elon 6404	. . . 4 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
4	1, 3	sylibr 234	. . 3 ⊢ (Lim 𝐴 → 𝐴 ∈ On)
5		cfval 10316	. . 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 6933	. . . 4 ⊢ (card‘𝑥) ∈ V
8	7	dfiin2 5057	. . 3 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9		df-rex 3077	. . . . . 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 3465	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
12		velpw 4627	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	12	anbi1i 623	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴))
14		coflim 10330	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝑥 = 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))
15	14	pm5.32da 578	. . . . . . . . . . 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 629	. . . . . . . 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 1920	. . . . . 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 2811	. . . 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 4975	. . 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 2793	. 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 2780	1 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  ∩ cint 4970  ∩ ciin 5016  Ord word 6394  Oncon0 6395  Lim wlim 6396  ‘cfv 6573  cardccrd 10004  cfccf 10006

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  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-ord 6398  df-on 6399  df-lim 6400  df-iota 6525  df-fun 6575  df-fv 6581  df-cf 10010

This theorem is referenced by:  cflim2  10332  cfss  10334  cfslb  10335