Theorem cflim3 10213

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 6402	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
2		cflim3.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	elon 6350	. . . 4 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
4	1, 3	sylibr 236	. . 3 ⊢ (Lim 𝐴 → 𝐴 ∈ On)
5		cfval 10197	. . 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 6875	. . . 4 ⊢ (card‘𝑥) ∈ V
8	7	dfiin2 4987	. . 3 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9		df-rex 3086	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)))
10		ancom 464	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}))
11		rabid 3434	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
12		velpw 4557	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	12	anbi1i 633	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴))
14		coflim 10212	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝑥 = 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))
15	14	pm5.32da 587	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
16	13, 15	bitrid 285	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
17	11, 16	bitrid 285	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
18	17	anbi2d 639	. . . . . . . 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 285	. . . . . . 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 1940	. . . . . 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 285	. . . . 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 2827	. . . 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 4907	. . 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 2809	. 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 2796	1 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085  {crab 3413  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4552  ∪ cuni 4862  ∩ cint 4902  ∩ ciin 4947  Ord word 6340  Oncon0 6341  Lim wlim 6342  ‘cfv 6516  cardccrd 9887  cfccf 9889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-ord 6344  df-on 6345  df-lim 6346  df-iota 6472  df-fun 6518  df-fv 6524  df-cf 9893

This theorem is referenced by:  cflim2  10214  cfss  10216  cfslb  10217